General Definitions

Abstract

A distance space (X, d) is a set X (carrier) equipped with a distance d.

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

1 Basic Definitions

• Distance

  A distance space (X, d) is a set X (carrier) equipped with a distance d.

  A function $d: X \times X \rightarrow \mathbb{R}$ is called a distance (or dissimilarity ) on X if, for all x, y ∈ X, it holds:

  • 1. d(x, y) ≥ 0 (nonnegativity);

  • 2. d(x, y) = d(y, x) (symmetry);

  • 3. d(x, x) = 0 (reflexivity).

  In Topology, a distance with d(x, y) = 0 implying x = y is called a symmetric .

  For any distance d, the function D ₁ defined for x ≠ y by D ₁(x, y) = d(x, y) + c, where c = max _{x, y, z ∈ X} (d(x, y) − d(x, z) − d(y, z)), and D(x, x) = 0, is a metric. Also, D ₂(x, y) = d(x, y)^c is a metric for sufficiently small c ≥ 0.

  The function D ₃(x, y) = inf∑ _i d(z _i , z _i+1), where the infimum is taken over all sequences $x = z_{0},\ldots,z_{n+1} = y$, is the path semimetric of the complete weighted graph on X, where, for any x, y ∈ X, the weight of edge xy is d(x, y).

• Similarity

  Let X be a set. A function $s: X \times X \rightarrow \mathbb{R}$ is called a similarity on X if s is nonnegative, symmetric and the inequality

  $$\displaystyle{s(x,y) \leq s(x,x)}$$

  holds for all x, y ∈ X, with equality if and only if x = y.

  The main transforms used to obtain a distance (dissimilarity) d from a similarity s bounded by 1 from above are: d = 1 − s, $d = \frac{1-s} {s}$, $d = \sqrt{1 - s}$, $d = \sqrt{2(1 - s^{2 } )}$, d = arccoss, d = −lns (cf. Chap. 4).

• Semimetric

  Let X be a set. A function $d: X \times X \rightarrow \mathbb{R}$ is called a semimetric on X if d is nonnegative, symmetric, reflexive (d(x, x) = 0 for x ∈ X) and it holds

  $$\displaystyle{d(x,y) \leq d(x,z) + d(z,y)}$$

  for all x, y, z ∈ X (triangle inequality or, sometimes, triangular inequality).

  In Topology, it is called a pseudo-metric (or, rarely, semidistance , gauge), while the term semimetric is sometimes used for a symmetric (a distance d(x, y) with d(x, y) = 0 only if x = y); cf. symmetrizable space in Chap. 2

  For a semimetric d, the triangle inequality is equivalent, for each fixed n ≥ 4 and all $x,y,z_{1},\ldots,z_{n-2} \in X$, to the following n-gon inequality

  $$\displaystyle{d(x,y) \leq d(x,z_{1}) + d(z_{1},z_{2}) +\ldots +d(z_{n-2},y).}$$

  Equivalent rectangle inequality is | d(x, y) − d(z ₁, z ₂) | ≤ d(x, z ₁) + +d(y, z ₂).

  For a semimetric d on X, define an equivalence relation, called metric identification , by x ∼ y if d(x, y) = 0; equivalent points are equidistant from all other points. Let [x] denote the equivalence class containing x; then D([x], [y]) = d(x, y) is a metric on the set {[x]: x ∈ X} of equivalence classes.

• Metric

  Let X be a set. A function $d: X \times X \rightarrow \mathbb{R}$ is called a metric on X if, for all x, y, z ∈ X, it holds:

  • 1. d(x, y) ≥ 0 (nonnegativity);

  • 2. d(x, y) = 0 if and only if x = y (identity of indiscernibles);

  • 3. d(x, y) = d(y, x) (symmetry);

  • 4. d(x, y) ≤ d(x, z) + d(z, y) (triangle inequality ).

  In fact, the above condition 1. follows from above 2., 3. and 4.

  If 2. is dropped, then d is called (Bukatin, 2002) relaxed semimetric. If 2. is weakened to “d(x, x) = d(x, y) = d(y, y) implies x = y”, then d is called relaxed metric . A partial metric is a partial semimetric, which is a relaxed metric.

  If above 2. is weakened to “d(x, y) = 0 implies x = y”, then d is called (Amini-Harandi, 2012) metric-like function . Any partial metric is metric-like.

• Metric space

  A metric space (X, d) is a set X equipped with a metric d.

  It is called a metric frame (or metric scheme, integral) if d is integer-valued.

  A pointed metric space (or rooted metric space) (X, d, x ₀) is a metric space (X, d) with a selected base point x ₀ ∈ X.

• Extended metric

  An extended metric is a generalization of the notion of metric: the value ∞ is allowed for a metric d.

• Quasi-distance

  Let X be a set. A function $d: X \times X \rightarrow \mathbb{R}$ is called a quasi-distance on X if d is nonnegative, and d(x, x) = 0 holds for all x ∈ X. It is also called a premetric or prametric in Topology and a divergence in Probability.

  If a quasi-distance d satisfies the strong triangle inequality d(x, y) ≤ d(x, z) + d(y, z), then (Lindenbaum, 1926) it is symmetric and so, a semimetric. A quasi-semimetric d is a semimetric if and only if (Weiss, 2012) it satisfies the full triangle inequality  | d(x, z) − d(z, y) | ≤ d(x, z) ≤ d(x, z) + d(z, y).

  The distance/metric notions are usually named as weakenings or modifications of the fundamental notion of metric, using various prefixes and modifiers. But, perhaps, extended (i.e., the value ∞ is allowed) semimetric and quasi-semimetric should be (as suggested in Lawvere, 2002) used as the basic terms, since, together with their short mappings, they are best behaved of the metric space categories.

• Quasi-semimetric

  A function $d: X \times X \rightarrow \mathbb{R}$ is called a quasi-semimetric (or hemimetric , ostensible metric) on X if d(x, x) = 0, d(x, y) ≥ 0 and the oriented triangle inequality

  $$\displaystyle{d(x,y) \leq d(x,z) + d(z,y)}$$

  holds for all x, y, z ∈ X. The set X can be partially ordered by the specialization order: x⪯y if and only if d(x, y) = 0.

  A weak quasi-metric is a quasi-semimetric d on X with weak symmetry, i.e., for all x, y ∈ X the equality d(x, y) = 0 implies d(y, x) = 0.

  An Albert quasi-metric is a quasi-semimetric d on X with weak definiteness, i.e., for all x, y ∈ X the equality d(x, y) = d(y, x) = 0 implies x = y.

  Both, weak and Albert, quasi-metric, is a usual quasi-metric.

  Any pre-order (X, ≺ ) (satisfying for all x, y, z ∈ X, x ≺ x and if x ≺ y and y ≺ z then x ≺ z) can be viewed as a pre-order extended quasi-semimetric (X, d) by defining d(x, y) = 0 if x ≺ y and d(x, y) = ∞, otherwise.

  A weightable quasi-semimetric is a quasi-semimetric d on X with relaxed symmetry, i.e., for all x, y, z ∈ X

  $$\displaystyle{d(x,y) + d(y,z) + d(z,x) = d(x,z) + d(z,y) + d(y,x),}$$

  holds or, equivalently, there exists a weight function $w(x) \in \mathbb{R}$ on X with d(x, y) − d(y, x) = w(y) − w(x) for all x, y ∈ X (i.e., $d(x,y) + \frac{1} {2}(w(x) - w(y))$ is a semimetric). If d is a weightable quasi-semimetric, then d(x, y) + w(x) is a partial semimetric (moreover, a partial metric if d is an Albert quasi-metric).

• Partial metric

  Let X be a set. A nonnegative symmetric function $p: X \times X \rightarrow \mathbb{R}$ is called a partial metric ([Matt92]) if, for all x, y, z ∈ X, it holds:

  • 1. p(x, x) ≤ p(x, y), i.e., every self-distance (or extent) p(x, x) is small;

  • 2. x = y if p(x, x) = p(x, y) = p(y, y) = 0 (T ₀ separation axiom);

  • 3. p(x, y) ≤ p(x, z) + p(z, y) − p(z, z) (sharp triangle inequality ).

  The 1-st above condition means that p is a forward resemblance, cf. Chap. 3

  If the 2-nd above condition is dropped, the function p is called a partial semimetric . The nonnegative function p is a partial semimetric if and only if p(x, y) − p(x, x) is a weightable quasi-semimetric with w(x) = p(x, x).

  If the 1-st above condition is also dropped, the function p is called (Heckmann, 1999) a weak partial semimetric . The nonnegative function p is a weak partial semimetric if and only if 2p(x, y) − p(x, x) − p(y, y) is a semimetric.

  Sometimes, the term partial metric is used when a metric d(x, y) is defined only on a subset of the set of all pairs x, y of points.

• Protometric

  A function $p: X \times X \rightarrow \mathbb{R}$ is called a protometric if, for all (equivalently, for all different) x, y, z ∈ X, the sharp triangle inequality holds:

  $$\displaystyle{p(x,y) \leq p(x,z) + p(z,y) - p(z,z).}$$

  For finite X, the matrix (( p(x, y))) is (Burkard et al., 1996) weak Monge array.

  A strong protometric is a protometric p with p(x, x) = 0 for all x ∈ X. Such a protometric is exactly a quasi-semimetric, but with the condition p(x, y) ≥ 0 (for any x, y ∈ X) being relaxed to p(x, y) + p(y, x) ≥ 0.

  A partial semimetric is a symmetric protometric (i.e., p(x, y) = p(y, x) with p(x, y) ≥ p(x, x) ≥ 0 for all x, y ∈ X.) An example of a nonpositive symmetric protometric is given by $p(x,y) = -(x.y)_{x_{0}} = \frac{1} {2}(d(x,y) - d(x,x_{0}) - d(y,y_{0}))$, where (X, d) is a metric space with a fixed base point x ₀ ∈ X; see Gromov product similarity $(x.y)_{x_{0}}$ and, in Chap. 4, Farris transform metric $C - (x.y)_{x_{0}}$.

  A 0-protometric is a protometric p for which all sharp triangle inequalities (equivalently, all inequalities p(x, y) + p(y, x) ≥ p(x, x) + p(y, y) implied by them) hold as equalities. For any u ∈ X, denote by A′ _u , A″ _u the 0-protometrics p with p(x, y) = 1 _x = u , 1 _y = u , respectively. The protometrics on X form a flat convex cone in which the 0-protometrics form the largest linear space. For finite X, a basis of this space is given by all but one A′ _u , A″ _u (since ∑ _u A′ _u  = ∑ _u A″ _u ) and, for the flat subcone of all symmetric 0-protometrics on X, by all A′ _u + A″ _u .

  A weighted protometric on X is a protometric with a point-weight function $w: X \rightarrow \mathbb{R}$. The mappings $p(x,y) = \frac{1} {2}(d(x,y) + w(x) + w(y))$ and d(x, y) = 2p(x, y) − p(x, x) − p(y, y), w(x) = p(x, x) establish a bijection between the weighted strong protometrics (d, w) and the protometrics p on X, as well as between the weighted semimetrics and the symmetric protometrics. For example, a weighted semimetric (d, w) with w(x) = −d(x, x ₀) corresponds to a protometric $-(x.y)_{x_{0}}$. For finite | X | , the above mappings amount to the representation

  $$\displaystyle{2p = d +\sum _{u\in X}p(u,u)(A'_{u} + A''_{u}).}$$

• Quasi-metric

  A function $d: X \times X \rightarrow \mathbb{R}$ is called a quasi-metric (or asymmetric metric, directed metric) on X if d(x, y) ≥ 0 holds for all x, y ∈ X with equality if and only if x = y, and for all x, y, z ∈ X the oriented triangle inequality

  $$\displaystyle{d(x,y) \leq d(x,z) + d(z,y)}$$

  holds. A quasi-metric space (X, d) is a set X equipped with a quasi-metric d.

  For any quasi-metric d, the functions max{d(x, y), d(y, x)} (called sometimes bi-distance), min{d(x, y), d(y, x)}, $\frac{1} {2}(d^{p}(x,y) + d^{p}(y,x))^{\frac{1} {p} }$ with given p ≥ 1 are metric generating; cf. Chap. 4.

  A non-Archimedean quasi-metric d is a quasi-distance on X which, for all x, y, z ∈ X, satisfies the following strengthened oriented triangle inequality:

  $$\displaystyle{d(x,y) \leq \max \{ d(x,z),d(z,y)\}.}$$

• Directed-metric

  Let X be a set. A function $d: X \times X \rightarrow \mathbb{R}$ is called (Jegede, 2005) a directed-metric on X if, for all x, y, z ∈ X, it holds d(x, y) = −d(y, x) and

  $$\displaystyle{\vert d(x,y)\vert \leq \vert d(x,z)\vert + \vert d(z,y)\vert.}$$

  Cf. displacement in Chap. 24 and rigid motion of metric space.

• Coarse-path metric

  Let X be a set. A metric d on X is called a coarse-path metric if, for a fixed C ≥ 0 and for every pair of points x, y ∈ X, there exists a sequence $x = x_{0},x_{1},\ldots,x_{t} = y$ for which d(x _i−1, x _i ) ≤ C for $i = 1,\ldots,t$, and it holds

  $$\displaystyle{d(x,y) \geq d(x_{0},x_{1}) + d(x_{1},x_{2}) +\ldots +d(x_{t-1},x_{t}) - C.}$$

• Near-metric

  Let X be a set. A distance d on X is called a near-metric (or C-near-metric) if d(x, y) > 0 for x ≠ y and the C-relaxed triangle inequality

  $$\displaystyle{d(x,y) \leq C(d(x,z) + d(z,y))}$$

  holds for all x, y, z ∈ X and some constant C ≥ 1.

  If d(x, y) > 0 for x ≠ y and the C-asymmetric triangle inequality d(x, y) ≤ d(x, z) + Cd(z, y) holds, d is a $\frac{C+1} {2}$-near-metric.

  A C-inframetric is a C-near-metric, while a C-near-metric is a 2C-inframetric.

  Some recent papers use the term quasi-triangle inequality for the above inequality and so, quasi-metric for the notion of near-metric.

  The power transform (Chap. 4) (d(x, y))^α of any near-metric is a near-metric for any α > 0. Also, any near-metric d admits a bi-Lipschitz mapping on (D(x, y))^α for some semimetric D on the same set and a positive number α.

  A near-metric d on X is called a Hölder near-metric if the inequality

  $$\displaystyle{\vert d(x,y) - d(x,z)\vert \leq \beta d^{\alpha }(y,z)(d(x,y) + d(x,z))^{1-\alpha }}$$

  holds for some β > 0, 0 < α ≤ 1 and all x, y, z ∈ X. Cf. Hölder mapping.

  A distance d on set X is said (Greenhoe, 2015) to satisfy (C, p) power triangle inequality if, for given positive C, p and any x, y, z ∈ X, it holds

  $$\displaystyle{d(x,y) \leq 2C\vert \frac{1} {2}d^{p}(x,z) + \frac{1} {2}d^{p}(z,y)\vert ^{\frac{1} {p} }.}$$

• f-quasi-metric

  Let $f(t,t'): \mathbb{R}_{\geq 0} \times \mathbb{R}_{\geq 0} \rightarrow \mathbb{R}_{\geq 0}$ be a function with lim_{(t, t′) → 0}, f(t, t′) = f(0, 0) = 0.

  Let X be a set. A function $d: X \times X \rightarrow \mathbb{R}$ is called (Arutyunov et al., 2016) a f-quasi-metric on X if d(x, y) ≥ 0 holds for all x, y ∈ X with equality if and only if x = y, and for all x, y, z ∈ X holds the f-triangle inequality

  $$\displaystyle{d(x,y) \leq f(d(x,z),d(z,y)).}$$

  The f-quasi-metric space (X, d) with symmetric d and f(t, t′) = max(t, t′) is exactly the Fréchet V -space (1906); cf. the partially ordered distance in Sect. 3.4

  The case f(t, t′) = t + t′ of a f-quasi-metric corresponds to a quasi-metric. Given q, q′ ≥ 1, the f-quasi-metric with f(t, t′) = qt + q′t′ is called (q,q′)-quasi-metric.

  The inequality d(x, y) ≤ F(d(x, z), d(y, z)) implies d(x, y) ≤ f(d(x, z), d(z, y)) for the function f(t, t′) = F(t, F(0, t′)).

• Weak ultrametric

  A weak ultrametric (or C-inframetric , C-pseudo-distance) d is a distance on X such that d(x, y) > 0 for x ≠ y and the C-inframetric inequality

  $$\displaystyle{d(x,y) \leq C\max \{d(x,z),d(z,y)\}}$$

  holds for all x, y, z ∈ X and some constant C ≥ 1.

  The term pseudo-distance is also used, in some applications, for any of a pseudo-metric, a quasi-distance, a near-metric, a distance which can be infinite, a distance with an error, etc. Another unsettled term is weak metric : it is used for both a near-metric and a quasi-semimetric.

• Ultrametric

  An ultrametric (or non-Archimedean metric) is (Krasner, 1944) a metric d on X which satisfies, for all x, y, z ∈ X, the following strengthened version of the triangle inequality (Hausdorff, 1934), called the ultrametric inequality :

  $$\displaystyle{d(x,y) \leq \max \{ d(x,z),d(z,y)\}}$$

  An ultrametric space is also called an isosceles space since at least two of d(x, y), d(z, y), d(x, z) are equal. An ultrametric on a set V has at most | V | values.

  A metric d is an ultrametric if and only if its power transform (see Chap. 4) d ^α is a metric for any real positive number α. Any ultrametric satisfies the four-point inequality. A metric d is an ultrametric if and only if it is a Farris transform metric (Chap. 4) of a four-point inequality metric.

• Robinsonian distance

  A distance d on X is called a Robinsonian distance (or monotone distance) if there exists a total order ⪯ on X compatible with it, i.e., for x, y, w, z ∈ X,

  $$\displaystyle{x\preceq y\preceq w\preceq z\mbox{ implies }d(y,w) \leq d(x,z),}$$

  or, equivalently, for x, y, z ∈ X, it holds

  $$\displaystyle{x\preceq y\preceq z\mbox{ implies }d(x,y) \leq \max \{ d(x,z),d(z,y)\}.}$$

  Any ultrametric is a Robinsonian distance.

• Four-point inequality metric

  A metric d on X is a four-point inequality metric (or additive metric ) if it satisfies the following strengthened version of the triangle inequality called the four-point inequality (Buneman, 1974): for all x, y, z, u ∈ X

  $$\displaystyle{d(x,y) + d(z,u) \leq \max \{ d(x,z) + d(y,u),d(x,u) + d(y,z)\}}$$

  holds. Equivalently, among the three sums d(x, y) + d(z, u), d(x, z) + d(y, u), d(x, u) + d(y, z) the two largest sums are equal.

  A metric satisfies the four-point inequality if and only if it is a tree-like metric.

  Any metric, satisfying the four-point inequality, is a Ptolemaic metric and an L ₁ -metric. Cf. L _p -metric in Chap. 5

  A bush metric is a metric for which all four-point inequalities are equalities, i.e., d(x, y) + d(u, z) = d(x, u) + d(y, z) holds for any u, x, y, z ∈ X.

• Relaxed four-point inequality metric

  A metric d on X satisfies the relaxed four-point inequality if, for all x, y, z, u ∈ X, among the three sums

  $$\displaystyle{d(x,y) + d(z,u),d(x,z) + d(y,u),d(x,u) + d(y,z)}$$

  at least two (not necessarily the two largest) are equal. A metric satisfies this inequality if and only if it is a relaxed tree-like metric.

• Ptolemaic metric

  A Ptolemaic metric d is a metric on X which satisfies the Ptolemaic inequality

  $$\displaystyle{d(x,y)d(u,z) \leq d(x,u)d(y,z) + d(x,z)d(y,u)}$$

  for all x, y, u, z ∈ X. A classical result, attributed to Ptolemy, says that this inequality holds in the Euclidean plane, with equality if and only if the points x, y, u, z lie on a circle in that order.

  A Ptolemaic space is a normed vector space (V, | | . | | ) such that its norm metric | | x − y | | is a Ptolemaic metric. A normed vector space is a Ptolemaic space if and only if it is an inner product space (Chap. 5); so, a Minkowskian metric (Chap. 6) is Euclidean if and only if it is Ptolemaic.

  For any metric d, the metric $\sqrt{d}$ is Ptolemaic ([FoSc06]).

• δ -hyperbolic metric

  Given a number δ ≥ 0, a metric d on a set X is called δ -hyperbolic if it satisfies the following Gromov δ-hyperbolic inequality (another weakening of the four-point inequality ): for all x, y, z, u ∈ X, it holds that

  $$\displaystyle{d(x,y) + d(z,u) \leq 2\delta +\max \{ d(x,z) + d(y,u),d(x,u) + d(y,z)\}.}$$

  A metric space (X, d) is δ-hyperbolic if and only if for all x ₀, x, y, z ∈ X it holds

  $$\displaystyle{(x.y)_{x_{0}} \geq \min \{ (x.z)_{x_{0}},(y.z)_{x_{0}}\}-\delta,}$$

  where $(x.y)_{x_{0}} = \frac{1} {2}(d(x_{0},x) + d(x_{0},y) - d(x,y))$ is the Gromov product of the points x and y of X with respect to the base point x ₀ ∈ X.

  A metric space (X, d) is 0-hyperbolic exactly when d satisfies the four-point inequality. Every bounded metric space of diameter D is D-hyperbolic. The n-dimensional hyperbolic space is ln3-hyperbolic.

  Every δ-hyperbolic metric space is isometrically embeddable into a geodesic metric space (Bonk and Schramm, 2000).

• Gromov product similarity

  Given a metric space (X, d) with a fixed point x ₀ ∈ X, the Gromov product similarity (or Gromov product, covariance, overlap function) $(.)_{x_{0}}$ is a similarity on X defined by

  $$\displaystyle{(x.y)_{x_{0}} = \frac{1} {2}(d(x,x_{0}) + d(y,x_{0}) - d(x,y)).}$$

  The triangle inequality for d implies $(x.y)_{x_{0}} \geq (x.z)_{x_{0}} + (y.z)_{x_{0}} - (z.z)_{x_{0}}$ (covariance triangle inequality ), i.e., sharp triangle inequality for protometric $-(x.y)_{x_{0}}$.

  If (X, d) is a tree, then $(x.y)_{x_{0}} = d(x_{0},[x,y])$. If (X, d) is a measure semimetric space, i.e., d(x, y) = μ(x△y) for a Borel measure μ on X, then (x. y)_∅ = μ(x ∩ y). If d is a distance of negative type, i.e., d(x, y) = d _E ²(x, y) for a subset X of a Euclidean space $\mathbb{E}^{n}$, then (x. y)₀ is the usual inner product on $\mathbb{E}^{n}$.

  Cf. Farris transform metric $d_{x_{0}}(x,y) = C - (x.y)_{x_{0}}$ in Chap. 4.

• Cross-difference

  Given a metric space (X, d) and quadruple (x, y, z, w) of its points, the cross-difference is the real number cd defined by

  $$\displaystyle{cd(x,y,z,w) = d(x,y) + d(z,w) - d(x,z) - d(y,w).}$$

  In terms of the Gromov product similarity, for all x, y, z, w, p ∈ X, it holds

  $$\displaystyle{\frac{1} {2}cd(x,y,z,w) = -(x.y)_{p} - (z.w)_{p} + (x.z)_{p} + (y.w)_{p};}$$

  in particular, it becomes (x. y) _p if y = w = p.

  If x ≠ z and y ≠ w, the cross-ratio is the positive number defined by

  $$\displaystyle{cr((x,y,z,w),d) = \frac{d(x,y)d(z,w)} {d(x,z)d(y,w)}.}$$

• 2k-gonal distance

  A 2k-gonal distance d is a distance on X which satisfies, for all distinct elements $x_{1},\ldots,x_{n} \in X$, the 2k-gonal inequality

  $$\displaystyle{\sum _{1\leq i<j\leq n}b_{i}b_{j}d(x_{i},x_{j}) \leq 0}$$

  for all $b \in \mathbb{Z}^{n}$ with ∑ _i = 1 ⁿ b _i  = 0 and ∑ _i = 1 ⁿ | b _i  |  = 2k.

• Distance of negative type

  A distance of negative type d is a distance on X which is 2k-gonal for any k ≥ 1, i.e., satisfies the negative type inequality

  $$\displaystyle{\sum _{1\leq i<j\leq n}b_{i}b_{j}d(x_{i},x_{j}) \leq 0}$$

  for all $b \in \mathbb{Z}^{n}$ with ∑ _i = 1 ⁿ b _i  = 0, and for all distinct elements $x_{1},\ldots,x_{n} \in X$.

  A distance can be of negative type without being a semimetric. Cayley proved that a metric d is an L ₂ -metric if and only if d ² is a distance of negative type.

• (2k + 1)-gonal distance

  A (2k + 1)-gonal distance d is a distance on X which satisfies, for all distinct elements $x_{1},\ldots,x_{n} \in X$, the (2k + 1)-gonal inequality

  $$\displaystyle{\sum _{1\leq i<j\leq n}b_{i}b_{j}d(x_{i},x_{j}) \leq 0}$$

  for all $b \in \mathbb{Z}^{n}$ with ∑ _i = 1 ⁿ b _i  = 1 and ∑ _i = 1 ⁿ | b _i  |  = 2k + 1.

  The (2k + 1)-gonal inequality with k = 1 is the usual triangle inequality. The (2k + 1)-gonal inequality implies the 2k-gonal inequality.

• Hypermetric

  A hypermetric d is a distance on X which is (2k + 1)-gonal for any k ≥ 1, i.e., satisfies the hypermetric inequality (Deza, 1960)

  $$\displaystyle{\sum _{1\leq i<j\leq n}b_{i}b_{j}d(x_{i},x_{j}) \leq 0}$$

  for all $b \in \mathbb{Z}^{n}$ with ∑ _i = 1 ⁿ b _i  = 1, and for all distinct elements $x_{1},\ldots,x_{n} \in X$.

  Any hypermetric is a semimetric, a distance of negative type and, moreover, it can be isometrically embedded into some n-sphere $\mathbb{S}^{n}$ with squared Euclidean distance. Any L ₁ -metric (cf. L _p -metric in Chap. 5) is a hypermetric.

• P-metric

  A P-metric d is a metric on X with values in [0, 1] which satisfies the correlation triangle inequality

  $$\displaystyle{d(x,y) \leq d(x,z) + d(z,y) - d(x,z)d(z,y).}$$

  The equivalent inequality 1 − d(x, y) ≥ (1 − d(x, z))(1 − d(z, y)) expresses that the probability, say, to reach x from y via z is either equal to (1 − d(x, z))(1 − d(z, y)) (independence of reaching z from x and y from z), or greater than it (positive correlation). A metric is a P-metric if and only if it is a Schoenberg transform metric (Chap. 4).

2 Main Distance-Related Notions

• Metric ball

  Given a metric space (X, d), the metric ball (or closed metric ball) with center x ₀ ∈ X and radius r > 0 is defined by $\overline{B}(x_{0},r) =\{ x \in X: d(x_{0},x) \leq r\}$, and the open metric ball with center x ₀ ∈ X and radius r > 0 is defined by B(x ₀, r) = { x ∈ X: d(x ₀, x) < r}. The closed ball is a subset of the closure of the open ball; it is a proper subset for, say, the discrete metric on X.

  The metric sphere with center x ₀ ∈ X and radius r > 0 is defined by S(x ₀, r) = { x ∈ X: d(x ₀, x) = r}.

  For the norm metric on an n-dimensional normed vector space (V, | | . | | ), the metric ball $\overline{B}^{n} =\{ x \in V: \vert \vert x\vert \vert \leq 1\}$ is called the unit ball, and the set S ⁿ⁻¹ = { x ∈ V:  | | x | |  = 1} is called the unit sphere. In a two-dimensional vector space, a metric ball (closed or open) is called a metric disk (closed or open, respectively).

• Metric hull

  Given a metric space (X, d), let M be a bounded subset of X.

  The metric hull H(M) of M is the intersection of all metric balls containing M.

  The set of surface points S(M) of M is the set of all x ∈ H(M) such that x lies on the sphere of one of the metric balls containing M.

• Distance-invariant metric space

  A metric space (X, d) is distance-invariant if all metric balls $\overline{B}(x_{0},r) =\{ x \in X: d(x_{0},x) \leq r\}$ of the same radius have the same number of elements.

  Then the growth rate of a metric space (X, d) is the function $f(n) = \vert \overline{B}(x,n)\vert$.

  (X, d) is a metric space of polynomial growth if there are some positive constants k, C such that f(n) ≤ Cn ^k for all n ≥ 0. Cf. graph of polynomial growth, including the group case, in Chap. 15

  For a metrically discrete metric space (X, d) (i.e., with a = inf _{x, y ∈ X, x ≠ y} d(x, y) > 0), its growth rate was defined also (Gordon–Linial–Rabinovich, 1998) by

  $$\displaystyle{\max _{x\in X,r\geq 2}\frac{\log \vert \overline{B}(x,ar)\vert } {\log r}.}$$

• Ahlfors q-regular metric space

  A metric space (X, d) endowed with a Borel measure μ is called Ahlfors q-regular if there exists a constant C ≥ 1 such that for every ball in (X, d) with radius r < diam(X, d) it holds

  $$\displaystyle{C^{-1}r^{q} \leq \mu (\overline{B}(x_{ 0},r)) \leq Cr^{q}.}$$

  If such an (X, d) is locally compact, then the Hausdorff q-measure can be taken as μ and q is the Hausdorff dimension. For two disjoint continua (nonempty connected compact metric subspaces ) C ₁, C ₂ of such space (X, d), let $\Gamma$ be the set of rectifiable curves connecting C ₁ to C ₂. The q-modulus between C ₁ and C ₂ is $M_{q}(C_{1},C_{2}) =\inf \{\int _{X}\rho ^{q}:\inf _{\gamma \in \Gamma }\int _{\gamma }\rho \geq 1\}$, where $\rho: X \rightarrow \mathbb{R}_{>0}$ is any density function on X; cf. the modulus metric in Chap. 6.

  The relative distance between C ₁ and C ₂ is $\delta (C_{1},C_{2}) = \frac{\inf \{d(\,p_{1},p_{2}):p_{1}\in C_{1},p_{2}\in C_{2}\}} {\min \{diam(C_{1}),diam(C_{2})\}}$. (X, d) is a q-Loewner space if there are increasing functions f, g: [0, ∞) → [0, ∞) such that for all C ₁, C ₂ it holds f(δ(C ₁, C ₂)) ≤ M _q (C ₁, C ₂) ≤ g(δ(C ₁, C ₂)).

• Connected metric space

  A metric space (X, d) is called connected if it cannot be partitioned into two nonempty open sets. Cf. connected space in Chap. 2

  The maximal connected subspaces of a metric space are called its connected components. A totally disconnected metric space is a space in which all connected subsets are ∅ and one-point sets.

  A path-connected metric space is a connected metric space such that any two its points can be joined by an arc (cf. metric curve ).

• Cantor connected metric space

  A metric space (X, d) is called Cantor (or pre-) connected if, for any two its points x, y and any ε > 0, there exists an ε-chain joining them, i.e., a sequence of points $x = z_{0},z_{1},\ldots,z_{n-1},z_{n} = y$ such that d(z _k , z _k+1) ≤ ε for every 0 ≤ k ≤ n. A metric space (X, d) is Cantor connected if and only if it cannot be partitioned into two remote parts A and B, i.e., such that inf{d(x, y): x ∈ A, y ∈ B} > 0.

  The maximal Cantor connected subspaces of a metric space are called its Cantor connected components. A totally Cantor disconnected metric is the metric of a metric space in which all Cantor connected components are one-point sets.

• Indivisible metric space

  A metric space (X, d) is called indivisible if it cannot be partitioned into two parts, neither of which contains an isometric copy of (X, d). Any indivisible metric space with | X | ≥ 2 is infinite, bounded and totally Cantor disconnected (Delhomme–Laflamme–Pouzet–Sauer, 2007).

  A metric space (X, d) is called an oscillation stable metric space (Nguyen Van Thé, 2006) if, given any ε > 0 and any partition of X into finitely many pieces, the ε -neighborhood of one of the pieces includes an isometric copy of (X, d).

• Closed subset of metric space

  Given a subset M of a metric space (X, d), a point x ∈ X is called a limit (or accumulation) point of M if any open metric ball B(x, r) = { y ∈ X: d(x, y) < r} contains a point x′ ∈ M with x′ ≠ x. The boundary ϑ(M) of M is the set of all its limit points. The closure of M, denoted by cl(M), is M ∪ϑ(M), and M is called closed subset, if M = cl(M), and dense subset, if X = cl(M).

  Every point of M which is not its limit point, is called an isolated point. The interior i n t(M) of M is the set of all its isolated points, and the exterior e x t(M) of M is int(X∖M). A subset M is called nowhere dense if int(cl(M)) = ∅.

  A subset M is called topologically discrete (cf. metrically discrete metric space ) if int(M) = M and dense-in-itself if int(M) = ∅. A dense-in-itself subset is called perfect (cf. perfect metric space ) if it is closed. The subsets Irr (irrational numbers) and $\mathbb{Q}$ (rational numbers) of $\mathbb{R}$ are dense, dense-in-itself but not perfect. The set $\mathbb{Q} \cap [0,1]$ is dense-in-itself but not dense in $\mathbb{R}$.

• Open subset of metric space

  A subset M of a metric space (X, d) is called open if, given any point x ∈ M, the open metric ball B(x, r) = { y ∈ X: d(x, y) < r} is contained in M for some number r > 0. The family of open subsets of a metric space forms a natural topology on it. A closed subset is the complement of an open subset.

  An open subset is called clopen, if it is closed, and a domain if it is connected.

  A door space is a metric (in general, topological) space in which every subset is either open or closed.

• Metric topology

  A metric topology is a topology induced by a metric; cf. equivalent metrics. More exactly, the metric topology on a metric space (X, d) is the set of all open sets of X, i.e., arbitrary unions of (finitely or infinitely many) open metric balls B(x, r) = { y ∈ X: d(x, y) < r}, x ∈ X, $r \in \mathbb{R}$, r > 0.

  A topological space which can arise in this way from a metric space is called a metrizable space (Chap. 2). Metrization theorems are theorems which give sufficient conditions for a topological space to be metrizable.

  On the other hand, the adjective metric in several important mathematical terms indicates connection to a measure, rather than distance, for example, metric Number Theory, metric Theory of Functions, metric transitivity.

• Equivalent metrics

  Two metrics d ₁ and d ₂ on a set X are called equivalent if they define the same topology on X, i.e., if, for every point x ₀ ∈ X, every open metric ball with center at x ₀ defined with respect to d ₁, contains an open metric ball with the same center but defined with respect to d ₂, and conversely.

  Two metrics d ₁ and d ₂ are equivalent if and only if, for every ε > 0 and every x ∈ X, there exists δ > 0 such that d ₁(x, y) ≤ δ implies d ₂(x, y) ≤ ε and, conversely, d ₂(x, y) ≤ δ implies d ₁(x, y) ≤ ε.

  All metrics on a finite set are equivalent; they generate the discrete topology.

• Metric betweenness

  The metric betweenness of a metric space (X, d) is (Menger, 1928) the set of all ordered triples (x, y, z) such that x, y, z are (not necessarily distinct) points of X for which the triangle equality d(x, y) + d(y, z) = d(x, z) holds.

• Monometric

  A ternary relation R on a set X is called a betweenness relation if (x, y, z) ∈ R if and only if (z, y, x) ∈ R and (x, y, z), (x, z, y) ∈ R if and only if y = z.

  Given a such relation R, a monometric is (Perez-Fernández et al., 2016) a function $d: X \times X \rightarrow \mathbb{R}_{\geq 0}$ with d(x, y) = 0 if and only if x = y and (x, y, z) implying d(x, y) ≤ d(x, z). Clearly, any metric is a monometric.

  Cf. a distance-rationalizable voting rule in Sect. 11.2

• Closed metric interval

  Given two different points x, y ∈ X of a metric space (X, d), the closed metric interval between them (or line induced by) them is the set of the points z, for which the triangle equality (or metric betweenness (x, z, y)) holds:

  $$\displaystyle{I(x,y) =\{ z \in X: d(x,y) = d(x,z) + d(z,y)\}.}$$

  Cf. inner product space (Chap. 5) and cutpoint additive metric (Chap. 15).

  Let Ext(x, y) = { z: y ∈ I(x, z)∖{x, z}}. A CC-line C C(x, y) is I(x, y) ∪ Ext(x, y) ∪ Ext(y, x). Chen–Chvátal, 2008, conjectured that every metric space on n, n ≥ 2, points, either has at least n distinct CC-lines or consists of a unique CC-line.

• Underlying graph of a metric space

  The underlying graph (or neighborhood graph) of a metric space (X, d) is a graph with the vertex-set X and xy being an edge if I(x, y) = { x, y}, i.e., there is no third point z ∈ X, for which d(x, y) = d(x, z) + d(z, y).

• Distance monotone metric space

  A metric space (X, d) is called distance monotone if for any its closed metric interval I(x, y) and u ∈ X∖I(x, y), there exists z ∈ I(x, xy) with d(u, z) > d(x, y).

• Metric triangle

  Three distinct points x, y, z ∈ X of a metric space (X, d) form a metric triangle if the closed metric intervals I(x, y), I(y, z) and I(z, x) intersect only in the common endpoints.

• Metric space having collinearity

  A metric space (X, d) has collinearity if for any ε > 0 each of its infinite subsets contains distinct ε-collinear (i.e., with d(x, y) + d(y, z) − d(x, z) ≤ ε) points x, y, z.

• Modular metric space

  A metric space (X, d) is called modular if, for any three different points x, y, z ∈ X, there exists a point u ∈ I(x, y) ∩ I(y, z) ∩ I(z, x). This should not be confused with modular distance in Chap. 10 and modulus metric in Chap. 6.

• Median metric space

  A metric space (X, d) is called a median metric space if, for any three points x, y, z ∈ X, there exists a unique point u ∈ I(x, y) ∩ I(y, z) ∩ I(z, x), or, equivalently,

  $$\displaystyle{d(x,u) + d(y,u) + d(z,u) = \frac{1} {2}((x,y) + d(y,z) + d(z,x)).}$$

  The point u is called median for {x,y,z}, since it minimises the sum of distances to them. Any median metric space is an L ₁ -metric; cf. L _p -metric in Chap. 5 and median graph in Chap. 15

  A metric space (X, d) is called an antimedian metric space if, for any three points x, y, z ∈ X, there exists a unique point u ∈ X maximizing d(x, u) + d(y, u) + d(z, u).

• Metric quadrangle

  Four different points x, y, z, u ∈ X of a metric space (X, d) form a metric quadrangle if x, z ∈ I(y, u) and y, u ∈ I(x, z); then d(x, y) = d(z, u) and d(x, u) = d(y, z).

  A metric space (X, d) is called weakly spherical if any three different points x, y, z ∈ X with y ∈ I(x, z), form a metric quadrangle with some point u ∈ X.

• Metric curve

  A metric curve (or, simply, curve) γ in a metric space (X, d) is a continuous mapping γ: I → X from an interval I of $\mathbb{R}$ into X. A curve is called an arc (or path, simple curve) if it is injective. A curve γ: [a, b] → X is called a Jordan curve (or simple closed curve) if it does not cross itself, and γ(a) = γ(b).

  The length of a curve γ: [a, b] → X is the number l(γ) defined by

  $$\displaystyle{l(\gamma ) =\sup \{\sum _{1\leq i\leq n}d(\gamma (t_{i}),\gamma (t_{i-1})): n \in \mathbb{N},a = t_{0} < t_{1} <\ldots < t_{n} = b\}.}$$

  A rectifiable curve is a curve with a finite length. A metric space (X, d), where every two points can be joined by a rectifiable curve, is called a quasi-convex metric space (or, specifically, C-quasi-convex metric space ) if there exists a constant C ≥ 1 such that every pair x, y ∈ X can be joined by a rectifiable curve of length at most Cd(x, y). If C = 1, then this length is equal to d(x, y), i.e., (X, d) is a geodesic metric space (Chap. 6).

  In a quasi-convex metric space (X, d), the infimum of the lengths of all rectifiable curves, connecting x, y ∈ X is called the internal metric.

  The metric d on X is called the intrinsic metric (and then (X, d) is called a length space ) if it coincides with the internal metric of (X, d).

  If, moreover, any pair x, y of points can be joined by a curve of length d(x, y), the metric d is called strictly intrinsic, and the length space (X, d) is a geodesic metric space. Hopf–Rinow, 1931, showed that any complete locally compact length space is geodesic and proper. The punctured plane $(\mathbb{R}^{2}\setminus \{0\},\vert \vert x - y\vert \vert _{2})$ is locally compact and path-connected but not geodesic: the distance between (−1, 0) and (1, 0) is 2 but there is no geodesic realizing this distance.

  The metric derivative of a metric curve γ: [a, b] → X at a limit point t is

  $$\displaystyle{\lim _{s\rightarrow 0}\frac{d(\gamma (t + s),\gamma (t))} {\vert s\vert },}$$

  if it exists. It is the rate of change, with respect to t, of the length of the curve at almost every point, i.e., a generalization of the notion of speed to metric spaces.

• Geodesic

  Given a metric space (X, d), a geodesic is a locally shortest metric curve, i.e., it is a locally isometric embedding of $\mathbb{R}$ into X; cf. Chap. 6.

  A subset S of X is called a geodesic segment (or metric segment , shortest path, minimizing geodesic) between two distinct points x and y in X, if there exists a segment (closed interval) [a,b] on the real line $\mathbb{R}$ and an isometric embedding γ: [a, b] → X, such that γ[a, b] = S, γ(a) = x and γ(b) = y.

  A metric straight line is a geodesic which is minimal between any two of its points; it is an isometric embedding of the whole of $\mathbb{R}$ into X. A metric ray and metric great circle are isometric embeddings of, respectively, the half-line $\mathbb{R}_{\geq 0}$ and a circle S ¹(0, r) into X.

  A geodesic metric space (Chap. 6) is a metric space in which any two points are joined by a geodesic segment. If, moreover, the geodesic is unique, the space is called totally geodesic (or uniquely geodesic).

  A geodesic metric space (X, d) is called geodesically complete if every geodesic is a subarc of a metric straight line. If (X, d) is complete, then it is geodesically complete. The punctured plane $(\mathbb{R}^{2}\setminus \{0\},\vert \vert x - y\vert \vert _{2})$ is not geodesically complete: any geodesic going to 0 is not a subarc of a metric straight line.

• Length spectrum

  Given a metric space (X, d), a closed geodesic is a map $\gamma: \mathbb{S}^{1} \rightarrow X$ which is locally minimizing around every point of $\mathbb{S}^{1}$.

  If (X, d) is a compact length space, its length spectrum is the collection of lengths of closed geodesics. Each length is counted with multiplicity equal to the number of distinct free homotopy classes that contain a closed geodesic of such length. The minimal length spectrum is the set of lengths of closed geodesics which are the shortest in their free homotopy class. Cf. the distance list.

• Systole of metric space

  Given a compact metric space (X, d), its systole sys(X, d) is the length of the shortest noncontractible loop in X; such a loop is a closed geodesic. So, sys(X, d) = 0 exactly if (X, d) is simply connected. Cf. connected space in Chap. 2

  If (X, d) is a graph with path metric, then its systole is referred to as the girth. If (X, d) is a closed surface, then its systolic ratio is the ratio $SR = \frac{sys^{2}(X,d)} {area(X,d)}$.

  Some tight upper bounds of SR for every metric on a surface are: $\frac{2} {\sqrt{3}} =\gamma _{2}$ (Hermite constant in 2D) for 2-torus (Loewner, 1949), $\frac{\pi }{2}$ for the real projective plane (Pu, 1952) and $\frac{\pi }{\sqrt{8}}$ for the Klein bottle (Bavard, 1986). Tight asymptotic bounds for a surface S of large genus g are $\frac{4} {9} \cdot \frac{\log ^{2}g} {\pi g} \leq SR(S) \leq \frac{\log ^{2}g} {\pi g}$ (Katz et al., 2007).

• Shankar–Sormani radii

  Given a geodesic metric space (X, d), Shankar and Sormani, 2009, defined its unique injectivity radius Uirad(X) as the supremum over all r ≥ 0 such that any two points at distance at most r are joined by a unique geodesic, and its minimal radius Mrad(X) as inf _p ∈ X d( p, MinCut( p)).

  Here the minimal cut locus of pMinCut( p) is the set of points q ∈ X for which there is a geodesic γ running from p to q such that γ extends past q but is not minimizing from p to any point past q. If (X, d) is a Riemannian space, then the distance function from p is a smooth function except at p itself and the cut locus. Cf. medial axis and skeleton in Chap. 21.

  It holds Uirad(X) ≤ Mrad(X) with equality if (X, d) is a Riemannian space in which case it is the injectivity radius. It holds Uirad(X) = ∞ for a flat disk but Mrad(X) < ∞ if (X, d) is compact and at least one geodesic is extendible.

• Geodesic convexity

  Given a geodesic metric space (X, d) and a subset M ⊂ X, the set M is called geodesically convex (or convex) if, for any two points of M, there exists a geodesic segment connecting them which lies entirely in M; the space is strongly convex if such a segment is unique and no other geodesic connecting those points lies entirely in M. The space is called locally convex if such a segment exists for any two sufficiently close points in M.

  For a given point x ∈ M, the radius of convexity is r _x  = sup{r ≥ 0: B(x, r) ⊂ M}, where the metric ball B(x, r) is convex. The point x is called the center of mass of points $y_{1},\ldots,y_{k} \in M$ if it minimizes the function ∑ _i d(x, y _i )² (cf. Fréchet mean ); such point is unique if d(y _i , y _j ) < r _x for all 1 ≤ i < j ≤ k.

  The injectivity radius of the set M is the supremum over all r ≥ 0 such that any two points in M at distance ≤ r are joined by unique geodesic segment which lies in M. The Hawaiian Earring is a compact complete metric space consisting of a set of circles of radius $\frac{1} {i}$ for each $i \in \mathbb{N}$ all joined at a common point; its injectivity radius is 0. It is path-connected but not simply connected.

  The set M ⊂ X is called a totally convex metric subspace of (X, d) if, for any two points of M, any geodesic segment connecting them lies entirely in M.

• Busemann convexity

  A geodesic metric space (X, d) is called Busemann convex (or Busemann space, nonpositively curved in the sense of Busemann) if, for any three points x, y, z ∈ X and midpoints m(x, z) and m(y, z) (i.e., $d(x,m(x,z)) = d(m(x,z),z) = \frac{1} {2}d(x,z)$ and $d(y,m(y,z)) = d(m(y,z),z) = \frac{1} {2}d(y,z)$), there holds

  $$\displaystyle{d(m(x,z),m(y,z)) \leq \frac{1} {2}d(x,y).}$$

  The flat Euclidean strip $\{(x,y) \in \mathbb{R}^{2}: 0 < x < 1\}$ is Gromov hyperbolic metric space (Chap. 6) but not Busemann convex one. In a complete Busemann convex metric space any two points are joined by a unique geodesic segment.

  A locally geodesic metric space (X, d) is called Busemann locally convex if the above inequality holds locally. Any locally CAT(0) metric space is Busemann locally convex.

• Menger convexity

  A metric space (X, d) is called Menger convex if, for any different points x, y ∈ X, there exists a third point z ∈ X for which d(x, y) = d(x, z) + d(z, y), i.e., | I(x, y) |  > 2 holds for the closed metric interval I(x, y) = { z ∈ X: (x, y) = d(x, z) + d(z, y)}. It is called strictly Menger convex if such a z is unique for all x, y ∈ X.

  Geodesic convexity implies Menger convexity. The converse holds for complete metric spaces.

  A subset M ⊂ X is called (Menger, 1928) a d-convex set (or interval-convex set) if I(x, y) ⊂ M for any different points x, y ∈ M. A function $f: M \rightarrow \mathbb{R}$ defined on a d-convex set M ⊂ X is a d-convex function if for any z ∈ I(x, y) ⊂  M

  $$\displaystyle{f(z) \leq \frac{d(y,z)} {d(x,y)}f(x) + \frac{d(x,z)} {d(x,y)}f(y).}$$

  A subset M ⊂ X is a gated set if for every x ∈ X there exists a unique x′ ∈ M, the gate, such that d(x, y) = d(x, x′) + d(x′, y) for y ∈ M. Any such set is d-convex.

• Midpoint convexity

  A metric space (X, d) is called midpoint convex (or having midpoints , admitting a midpoint map) if, for any different points x, y ∈ X, there exists a third point m(x, y) ∈ X for which $d(x,m(x,y)) = d(m(x,y),y) = \frac{1} {2}d(x,y)$. Such a point m(x, y) is called a midpoint and the map m: X × X → X is called a midpoint map (cf. midset ); this map is unique if m(x, y) is unique for all x, y ∈ X.

  For example, the geometric mean $\sqrt{xy}$ is the midpoint map for the metric space $(\mathbb{R}_{>0},d(x,y) = \vert \log x -\log y\vert )$.

  A complete metric space is geodesic if and only if it is midpoint convex.

  A metric space (X, d) is said to have approximate midpoints if, for any points x, y ∈ X and any ε > 0, there exists an ε-midpoint, i.e., a point z ∈ X such that $d(x,z) \leq \frac{1} {2}d(x,y)+\epsilon \geq d(z,y)$.

• Ball convexity

  A midpoint convex metric space (X, d) is called ball convex if

  $$\displaystyle{d(m(x,y),z) \leq \max \{ d(x,z),d(y,z)\}}$$

  for all x, y, z ∈ X and any midpoint map m(x, y).

  Ball convexity implies that all metric balls are totally convex and, in the case of a geodesic metric space, vice versa. Ball convexity implies also the uniqueness of a midpoint map (geodesics in the case of complete metric space).

  The metric space $(\mathbb{R}^{2},d(x,y) =\sum _{ i=1}^{2}\sqrt{\vert x_{i } - y_{i } \vert })$ is not ball convex.

• Distance convexity

  A midpoint convex metric space (X, d) is called distance convex if

  $$\displaystyle{d(m(x,y),z) \leq \frac{1} {2}(d(x,z) + d(y,z)).}$$

  A geodesic metric space is distance convex if and only if the restriction of the distance function d(x, ⋅ ), x ∈ X, to every geodesic segment is a convex function.

  Distance convexity implies ball convexity and, in the case of Busemann convex metric space, vice versa.

• Metric convexity

  A metric space (X, d) is called metrically convex if, for any different points x, y ∈ X and any λ ∈ (0, 1), there exists a third point z = z(x, y, λ) ∈ X for which d(x, y) = d(x, z) + d(z, y) and d(x, z) = λ d(x, y).

  The space is called strictly metrically convex if such a point z(x, y, λ) is unique for all x, y ∈ X and any λ ∈ (0, 1).

  A metric space (X, d) is called strongly metrically convex if, for any different points x, y ∈ X and any λ ₁, λ ₂ ∈ (0, 1), there exists a third point z = z(x, y, λ) ∈ X for which d(z(x, y, λ ₁), z(x, y, λ ₂)) =  | λ ₁ −λ ₂ | d(x, y).

  Metric convexity implies Menger convexity, and every Menger convex complete metric space is strongly metrically convex.

  A metric space (X, d) is called nearly convex (Mandelkern, 1983) if, for any different points x, y ∈ X and any λ, μ > 0 such that d(x, y) < λ +μ, there exists a third point z ∈ X for which d(x, z) < λ and d(z, y) < μ, i.e., z ∈ B(x, λ) ∩ B(y, μ). Metric convexity implies near convexity.

• Takahashi convexity

  A metric space (X, d) is called Takahashi convex if, for any different points x, y ∈ X and any λ ∈ (0, 1), there exists a third point z = z(x, y, λ) ∈ X such that d(z(x, y, λ), u) ≤ λ d(x, u) + (1 −λ)d(y, u) for all u ∈ X. Any convex subset of a normed space is a Takahashi convex metric space with z(x, y, λ) = λ x + (1 −λ)y.

  A set M ⊂ X is Takahashi convex if z(x, y, λ) ∈ M for all x, y ∈ X and any λ ∈ [0, 1]. In a Takahashi convex metric space, all metric balls, open metric balls, and arbitrary intersections of Takahashi convex subsets are all Takahashi convex.

• Hyperconvexity

  A metric space (X, d) is called hyperconvex (Aronszajn–Panitchpakdi, 1956) if it is metrically convex and its metric balls have the infinite Helly property, i.e., any family of mutually intersecting closed balls in X has nonempty intersection. A metric space (X, d) is hyperconvex if and only if it is an injective metric space.

  The spaces l _∞ ⁿ, l _∞ ^∞ and l ₁ ² are hyperconvex but l ₂ ^∞ is not.

• Distance matrix

  Given a finite metric space (X = { x ₁, ⋯ , x _n }, d), its distance matrix is the symmetric n × n matrix ((d _ij )), where d _ij  = d(x _i , x _j ) for any 1 ≤ i, j ≤ n.

  The probability that a symmetric n × n matrix, whose diagonal elements are zeros and all other elements are uniformly random real numbers, is a distance matrix is (Mascioni, 2005) $\frac{1} {2}$, $\frac{17} {120}$ for n = 3, 4, respectively.

• Magnitude of a finite metric space

  Let $(X =\{ x_{1},\ldots,x_{n}\},d)$ be a finite metric space, such that there exists a vector $w =\{ w_{1},\ldots,w_{n}\}$ with $((e^{-d(x_{i},x_{j})}))w = (1,\ldots,1)^{T}$.

  Then the magnitude of (X, d) is (Leinster–Meckes, 2016) the sum ∑ _i = 1 ⁿ w _i . In fact, the definition of Euler characteristic of a category was generalized to enriched categories, renamed magnitude, then re-specialized to metric spaces.

• Distance product of matrices

  Given n × n matrices A = ((a _ij )) and B = ((b _ij )), their distance (or min-plus) product is the n × n matrix C = ((c _ij )) with c _ij  = min _k = 1 ⁿ(a _ik + b _kj ).

  It is the usual matrix multiplication in the tropical semiring $(\mathbb{R} \cup \{\infty \},\min,+)$ (Chap. 18). If A is the matrix of weights of an edge-weighted complete graph K _n , then its direct power A ⁿ is the (shortest path) distance matrix of this graph.

• Distance list

  Given a metric space (X, d), its distance set and distance list are the set and the multiset (i.e., multiplicities are counted) and of all pairwise distances.

  Two subsets A, B ⊂ X are said to be homometric sets if they have the same distance list. Cf. homometric structures in Chap. 24.

  A finite metric space is called tie-breaking if all pairwise distances are distinct.

• Degree of distance near-equality

  Given a finite metric space (X, d) with | X |  = n ≥ 3, let $f =\min \vert \frac{d(x,y)} {d(a,b)} - 1\vert$ (degree of distance near-equality ) and $f' =\min \vert \frac{d(x,y)} {d(x,b)} - 1\vert$, where the minimum is over different 2-subsets {x, y}, {a, b} of X and, respectively, over different x, y, b ∈ X. [OpPi14] proved $f \leq \frac{9\log n} {n^{2}}$ and $f' \leq \frac{3} {n}$, while $f \geq \frac{\log n} {20n^{2}}$ and $f' \geq \frac{1} {2n}$ for some (X, d).

• Semimetric cone

  The semimetric cone MET _n is the polyhedral cone in $\mathbb{R}^{{n\choose 2}}$ of all distance matrices of semimetrics on the set V _n  = { 1, …, n}. Vershik, 2004, considers MET _∞ , i.e., the weakly closed convex cone of infinite distance matrices of semimetrics on $\mathbb{N}$.

  The cone of n-point weightable quasi-semimetrics is a projection along an extreme ray of the semimetric cone Met _n+1 (Grishukhin–Deza–Deza, 2011).

  The metric fan is a canonical decomposition MF _n of MET _n into subcones whose faces belong to the fan, and the intersection of any two of them is their common boundary. Two semimetrics d, d′ ∈ MET _n lie in the same cone of the metric fan if the subdivisions δ _d , δ _d′ of the polyhedron $\delta (n,2) = conv\{e_{i} + e_{j}: 1 \leq i < j \leq n\} \subset \mathbb{R}^{n}$ are equal. Here a subpolytope P of δ(n, 2) is a cell of the subdivision δ _d if there exists $y \in \mathbb{R}^{n}$ satisfying y _i + y _j  = d _ij if e _i + e _j is a vertex of P, and y _i + y _j  > d _ij , otherwise. The complex of bounded faces of the polyhedron dual to δ _d is the tight span of the semimetric d.

• Cayley–Menger matrix

  Given a finite metric space (X = { x ₁, ⋯ , x _n }, d), its Cayley–Menger matrix is the symmetric (n + 1) × (n + 1) matrix

  $$\displaystyle{CM(X,d) = \left (\begin{array}{cc} 0 & e\\ e^{T } &D\end{array} \right ),}$$

  where D = ((d ²(x _i , x _j ))) and e is the n-vector all components of which are 1.

  The determinant of CM(X, d) is called the Cayley–Menger determinant. If (X, d) is a metric subspace of the Euclidean space $\mathbb{E}^{n-1}$, then CM(X, d) is (−1)ⁿ2ⁿ⁻¹((n − 1)! )² times the squared (n − 1)-dimensional volume of the convex hull of X in $\mathbb{R}^{n-1}$.

• Gram matrix

  Given elements $v_{1},\ldots,v_{k}$ of a Euclidean space, their Gram matrix is the symmetric k × k matrix VV ^T, where V = ((v _ij )), of pairwise inner products of $v_{1},\ldots,v_{k}$:

  $$\displaystyle{G(v_{1},\ldots,v_{k}) = ((\langle v_{i},v_{j}\rangle )).}$$

  It holds $G(v_{1},\ldots,v_{k}) = \frac{1} {2}((d_{E}^{2}(v_{ 0},v_{i}) + d_{E}^{2}(v_{ 0},v_{j}) - d_{E}^{2}(v_{ i},v_{j})))$, i.e., the inner product 〈⋅ , ⋅ 〉 is the Gromov product similarity of the squared Euclidean distance d _E ². A k × k matrix ((d _E ²(v _i , v _j ))) is called Euclidean distance matrix (or EDM). It defines a distance of negative type on $\{1,\ldots,k\}$; all such matrices form the (nonpolyhedral) closed convex cone of all such distances.

  The determinant of a Gram matrix is called the Gram determinant; it is equal to the square of the k-dimensional volume of the parallelotope constructed on $v_{1},\ldots v_{k}$.

  A symmetric k × k real matrix M is said to be positive-semidefinite (PSD) if xMx ^T ≥ 0 for any nonzero $x \in \mathbb{R}^{k}$ and positive-definite (PD) if xMx ^T > 0. A matrix is PSD if and only if it is a Gram matrix; it is PD if and only the vectors $v_{1},\ldots,v_{k}$ are linearly independent. In Statistics, the covariance matrices and correlation matrices are exactly PSD and PD ones, respectively.

• Midset

  Given a metric space (X, d) and distinct y, z ∈ X, the midset (or bisector) of points y and z is the set M = { x ∈ X: d(x, y) = d(x, z)} of midpoints x.

  A metric space is said to have the n-point midset property if, for every pair of its points, the midset has exactly n points. The one-point midset property means uniqueness of the midpoint map. Cf. midpoint convexity.

• Distance k-sector

  Given a metric space (X, d) and disjoint subsets Y, Z ⊂ X, the bisector of Y and Z is the set M = { x ∈ X: inf _y ∈ Y d(x, y) = inf _z ∈ Z d(x, z)}.

  The distance k-sector of Y and Z is the sequence $M_{1},\ldots,M_{k-1}$ of subsets of X such that M _i , for any 1 ≤ i ≤ k − 1, is the bisector of sets M _i−1 and M _i+1, where Y = M ₀ and Z = M _k . Asano–Matousek–Tokuyama, 2006, considered the distance k-sector on the Euclidean plane $(\mathbb{R}^{2},l_{2})$; for compact sets Y and Z, the sets $M_{1},\ldots,M_{k-1}$ are curves partitioning the plane into k parts.

• Metric basis

  Given a metric space (X, d) and a subset M ⊂ X, for any point x ∈ X, its metric M-representation is the set {(m, d(x, m)): m ∈ M} of its metric M-coordinates (m, d(x, m)). The set M is called (Blumenthal, 1953) a metric basis (or resolving set, locating set, set of uniqueness, set of landmarks) if distinct points x ∈ X have distinct M-representations. A vertex-subset M of a connected graph is (Okamoto et al., 2009) a local metric basis if adjacent vertices have distinct M-representations.

  The resolving number of a finite (X, d) is (Chartrand–Poisson–Zhang, 2000) minimum k such that any k-subset of X is a metric basis.

  The vertices of a non degenerate simplex form a metric basis of $\mathbb{E}^{n}$, but l ₁- and l _∞ -metrics on $\mathbb{R}^{n}$, n > 1, have no finite metric basis.

  The distance similarity is (Saenpholphat–Zhang, 2003) an equivalence relation on X defined by x ∼ y if d(z, x) = d(z, y) for any z ∈ X∖{x, y}. Any metric basis contains all or all but one elements from each equivalence class.

3 Metric Numerical Invariants

• Resolving dimension

  Given a metric space (X, d), its resolving dimension (or location number (Slater, 1975), metric dimension (Harary–Melter, 1976)) is the minimum cardinality of its metric basis. The upper resolving dimension of (X, d) is the maximum cardinality of its metric basis not containing another metric basis as a proper subset. Adjacency dimension of (X, d) is the metric dimension of (X, min(2, d)).

  A metric independence number of (X, d) is (Currie–Oellermann, 2001) the maximum cardinality I of a collection of pairs of points of X, such that for any two, (say, (x, y) and (x′, y′)) of them there is no point z ∈ X with d(z, x) ≠ d(z, y) and d(z, x′) ≠ d(z, y′). A function f: X → [0, 1] is a resolving function of (X, d) if ∑ _{z ∈ X: d(x, z) ≠ d(y, z)} f(z) ≥ 1 for any distinct x, y ∈ X. The fractional resolving dimension of (X, d) is F = min∑ _x ∈ X g(x), where the minimum is taken over resolving functions f such that any function f′ with f′, f is not resolving.

  The partition dimension of (X, d) is (Chartrand–Salevi–Zhang, 1998) the minimum cardinality P of its resolving partition, i.e., a partition X = ∪_{1 ≤ i ≤ k} S _i such that no two points have, for 1 ≤ i ≤ k, the same minimal distances to the set S _i .

  Related locating a robber game on a graph G = (V, E) was considered in 2012 by Seager and by Carraher et al.: cop win on G if every sequence $r = r_{1},\ldots,r_{n}$ of robber’s steps (r _i  ∈ V and d _path(r _i , r _i+1) ≤ 1) is uniquely identified by a sequence $d(r_{1},c_{1}),\ldots,d(r_{n},c_{n})$ of cop’s distance queries for some $c_{1},\ldots,c_{n} \in V$.

• Metric dimension

  For a metric space (X, d) and a number ε > 0, let C _ε be the minimal size of an ε -net of (X, d), i.e., a subset M ⊂ X with ∪ _x ∈ M B(x, ε) = X. The number

  $$\displaystyle{dim(X,d) =\lim _{\epsilon \rightarrow 0}\frac{\ln C_{\epsilon }} {-\ln \epsilon }}$$

  (if it exists) is called the metric dimension (or Minkowski–Bouligand dimension, box-counting dimension ) of X. If the limit above does not exist, then the following notions of dimension are considered:

  • 1. $\underline{dim}(X,d) =\underline{\lim } _{\epsilon \rightarrow 0}\frac{\ln C_{\epsilon }} {-\ln \epsilon }$ called the lower Minkowski dimension (or lower dimension, lower box dimension, Pontryagin–Snirelman dimension);

  • 2. $\overline{dim}(X,d) = \overline{\lim }_{\epsilon \rightarrow 0}\frac{\ln C_{\epsilon }} {-\ln \epsilon }$ called the Kolmogorov–Tikhomirov dimension (or upper dimension, entropy dimension, upper box dimension).

  See below examples of other, less prominent, notions of metric dimension.

  • 1. The (equilateral) metric dimension of a metric space is the maximum cardinality of its equidistant subset, i.e., such that any two of its distinct points are at the same distance. For a normed space, this dimension is equal to the maximum number of translates of its unit ball that touch pairwise.

  • 2. For any c > 1, the (normed space) metric dimension d i m _c (X) of a finite metric space (X, d) is the least dimension of a real normed space (V, | | . | | ) such that there is an embedding f: X → V with $\frac{1} {c}d(x,y) \leq \vert \vert \,f(x) - f(y)\vert \vert \leq d(x,y)$.

  • 3. The (Euclidean) metric dimension of a finite metric space (X, d) is the least dimension n of a Euclidean space $\mathbb{E}^{n}$ such that (X, f(d)) is its metric subspace, where the minimum is taken over all continuous monotone increasing functions f(t) of t ≥ 0.

  • 4. The dimensionality of a metric space is $\frac{\mu ^{2}} {2\sigma ^{2}}$, where μ and σ ² are the mean and variance of its histogram of distance values; this notion is used in Information Retrieval for proximity searching.

    The term dimensionality is also used for the minimal dimension, if it is finite, of Euclidean space in which a given metric space embeds isometrically.

• Hausdorff dimension

  Given a metric space (X, d) and p, q > 0, let H _p ^q = inf∑ _i = 1 ^∞(diam(A _i ))^p, where the infimum is taken over all countable coverings {A _i } with diameter of A _i less than q. The Hausdorff q-measure of X is the metric outer measure defined by

  $$\displaystyle{H^{p} =\lim _{ q\rightarrow 0}H_{p}^{q}.}$$

  The Hausdorff dimension (or fractal dimension ) of (X, d) is defined by

  $$\displaystyle{dim_{Haus}(X,d) =\inf \{ p \geq 0: H^{p}(X) = 0\}.}$$

  Any countable metric space has dim _Haus  = 0, $dim_{Haus}(\mathbb{E}^{n}) = n$, and any $X \subset \mathbb{E}^{n}$ with Int X ≠ ∅ has $dim_{Haus} = \overline{dim}$. For any totally bounded (X, d), it holds

  $$\displaystyle{dim_{top} \leq dim_{Haus} \leq \underline{ dim} \leq dim \leq \overline{dim}.}$$

• Rough dimension

  Given a metric space (X, d), its rough n-volume V o l _n X is $\overline{\lim }_{\epsilon \rightarrow 0}\epsilon ^{n}\beta _{X}(\epsilon )$, where ε > 0 and β _X (ε) = max | Y | for Y ⊆ X with d(a, b) ≥ ε if a ∈ Y, b ∈ Y∖{a}; β _X (ε) = ∞ is permitted. The rough dimension is defined ([BBI01]) by

  $$\displaystyle{dim_{rough}(X,d) =\sup \{ n: V ol_{n}X = \infty \}\mbox{ or, equivalently, } =\inf \{ n: V ol_{n}X = 0\}.}$$

  The space (X, d) can be not locally compact. It holds dim _Haus  ≤ dim _rough .

• Packing dimension

  Given a metric space (X, d) and p, q > 0, let P _p ^q = sup∑ _i = 1 ^∞(diam(B _i ))^p, where the supremum is taken over all countable packings (by disjoint balls) {B _i } with the diameter of B _i less than q.

  The packing q-pre-measure is P ₀ ^p = lim _q → 0 P _p ^q. The packing q-measure is the metric outer measure which is the infimum of packing q-pre-measures of countable coverings of X. The packing dimension of (X, d) is defined by

  $$\displaystyle{dim_{pack}(X,d) =\inf \{ p \geq 0: P^{p}(X) = 0\}.}$$

• Topological dimension

  For any compact metric space (X, d) its topological dimension (or Lebesgue covering dimension ) is defined by

  $$\displaystyle{\dim _{top}(X,d) =\inf _{d'}\{\dim _{Haus}(X,d')\},}$$

  where d′ is any metric on X equivalent to d. So, it holds dim _top  ≤ dim _Haus . A fractal (Chap. 18) is a metric space for which this inequality is strict.

  This dimension does not exceed also the Assouad–Nagata dimension of (X, d).

  In general, the topological dimension of a topological space X is the smallest integer n such that, for any finite open covering of X, there exists a finite open refinement of it with no point of X belonging to more than n + 1 elements.

  The geometric dimension is (Kleiner, 1999; [BBI01]) supdim _top (Y, d) over compact Y ⊂ X.

• Doubling dimension

  The doubling dimension (dim _doubl (X, d)) of a metric space (X, d) is the smallest integer n (or ∞ if such an n does not exist) such that every metric ball (or, say, a set of finite diameter) can be covered by a family of at most 2ⁿ metric balls (respectively, sets) of half the diameter.

  If (X, d) has finite doubling dimension, then d is called a doubling metric and the smallest integer m such that every metric ball can be covered by a family of at most m metric balls of half the diameter is called doubling constant.

• Assouad–Nagata dimension

  The Assouad–Nagata dimension dim _AN (X, d) of a metric space (X, d) is the smallest integer n (or ∞ if such an n does not exist) for which there exists a constant C > 0 such that, for all s > 0, there exists a covering of X by its subsets of diameter ≤ Cs with every subset of X of diameter ≤ s meeting ≤ n + 1 elements of covering. It holds (LeDonne–Rajala, 2014) dim _AN  ≤ dim _doubl ; but dim _AN  = 1, while dim _doubl  = ∞, holds (Lang–Schlichenmaier, 2014) for some real trees (X, d).

  Replacing “for all s > 0” in the above definition by “for s > 0 sufficiently large” or by “for s > 0 sufficiently small”, gives the microscopic m i-dim _AN (X, d) and macroscopic m a-dim _AN (X, d) Assouad–Nagata dimensions, respectively. Then (Brodskiy et al., 2006) mi-dim _AN (X, d) = dim _AN (X, min{d, 1}) and

  ma-dim _AN (X, d) = dim _AN (X, max{d, 1}) (here max{d(x, y), 1} means 0 for x = y).

  The Assouad–Nagata dimension is preserved (Lang–Schlichenmaier, 2004) under quasi-symmetric mapping but, in general, not under quasi-isometry.

• Vol’berg–Konyagin dimension

  The Vol’berg–Konyagin dimension of a metric space (X, d) is the smallest constant C > 1 (or ∞ if such a C does not exist) for which X carries a doubling measure, i.e., a Borel measure μ such that, for all x ∈ X and r > 0, it holds that

  $$\displaystyle{\mu (\overline{B}(x,2r)) \leq C\mu (\overline{B}(x,r)).}$$

  A metric space (X, d) carries a doubling measure if and only if d is a doubling metric, and any complete doubling metric carries a doubling measure.

  The Karger–Ruhl constant of a metric space (X, d) is the smallest c > 1 (or ∞ if such a c does not exist) such that for all x ∈ X and r > 0 it holds

  $$\displaystyle{\vert \overline{B}(x,2r)\vert \leq c\vert \overline{B}(x,r)\vert.}$$

  If c is finite, then the doubling dimension of (X, d) is at most 4c.

• Hyperbolic dimension

  A metric space (X, d) is called an (R,N)-large-scale doubling if there exists a number R > 0 and integer N > 0 such that every ball of radius r ≥ R in (X, d) can be covered by N balls of radius $\frac{r} {2}$.

  The hyperbolic dimension hypdim(X, d) of a metric space (X, d) (Buyalo–Schroeder, 2004) is the smallest integer n such that for every r > 0 there are R > 0, an integer N > 0 and a covering of X with the following properties:

  • 1. Every ball of radius r meets at most n + 1 elements of the covering;

  • 2. The covering is an (R, N)-large-scale doubling, and any finite union of its elements is an (R′, N)-large-scale doubling for some R′ > 0.

  The hyperbolic dimension is 0 if (X, d) is a large-scale doubling, and it is n if (X, d) is n-dimensional hyperbolic space.

  Also, hypdim(X, d) ≤ asdim(X, d) since the asymptotic dimension asdim(X, d) corresponds to the case N = 1 in the definition of hypdim(X, d).

  The hyperbolic dimension is preserved under a quasi-isometry.

• Asymptotic dimension

  The asymptotic dimension asdim(X, d) of a metric space (X, d) (Gromov, 1993) is the smallest integer n such that, for every r > 0, there exists a constant D = D(r) and a covering of X by its subsets of diameter at most D such that every ball of radius r meets at most n + 1 elements of the covering.

  The asymptotic dimension is preserved under a quasi-isometry.

• Width dimension

  Let (X, d) be a compact metric space. For a given number ε > 0, the width dimension Widim _ε (X, d) of (X, d) is (Gromov, 1999) the minimum integer n such that there exists an n-dimensional polyhedron P and a continuous map f: X → P (called an ε-embedding) with diam( f ⁻¹(y)) ≤ ε for all y ∈ P.

  The width dimension is a macroscopic dimension at the scale ≥ε of (X, d), because its limit for ε → 0 is the topological dimension of (X, d).

• Godsil–McKay dimension

  We say that a metric space (X, d) has Godsil–McKay dimension n ≥ 0 if there exists an element x ₀ ∈ X and two positive constants c and C such that the inequality ck ⁿ ≤ | {x ∈ X: d(x, x ₀) ≤ k} | ≤ Ck ⁿ holds for every integer k ≥ 0.

  This notion was introduced in [GoMc80] for the path metric of a countable locally finite graph. They proved that, if the group $\mathbb{Z}^{n}$ acts faithfully and with a finite number of orbits on the vertices of the graph, then this dimension is n.

• Metric outer measure

  A σ-algebra over X is any nonempty collection $\Sigma$ of subsets of X, including X itself, that is closed under complementation and countable unions of its members.

  Given a σ-algebra $\Sigma$ over X, a measure on $(X,\Sigma )$ is a function $\mu: \Sigma \rightarrow [0,\infty ]$ with the following properties:

  • 1. μ(∅) = 0;

  • 2. For any sequence {A _i } of pairwise disjoint subsets of X, μ(∑ _i A _i ) = ∑ _i μ(A _i ) (countable σ-additivity).

  The triple $(X,\Sigma,\mu$) is called a measure space. If $M \subset A \in \Sigma$ and μ(A) = 0 implies $M \in \Sigma$, then $(X,\Sigma,\mu$) is called a complete measure space. A measure μ with μ(X) = 1 is called a probability measure.

  If X is a topological space (see Chap. 2), then the σ-algebra over X, consisting of all open and closed sets of X, is called the Borel σ-algebra of X, $(X,\Sigma )$ is called a Borel space, and a measure on $\Sigma$ is called a Borel measure. So, any metric space (X, d) admits a Borel measure coming from its metric topology, where the open set is an arbitrary union of open metric d-balls.

  An outer measure on X is a function ν: P(X) → [0, ∞] (where P(X) is the set of all subsets of X) with the following properties:

  • 1. ν(∅) = 0;

  • 2. For any subsets A, B ⊂ X, A ⊂ B implies ν(A) ≤ ν(B) (monotonicity);

  • 3. For any sequence {A _i } of subsets of X, ν(∑ _i A _i ) ≤ ∑ _i ν(A _i ) (countable subadditivity).

  A subset M ⊂ X is called ν-measurable if ν(A) = ν(A ∪ M) +ν(A∖M) for any A ⊂ X. The set $\Sigma '$ of all ν-measurable sets forms a σ-algebra over X, and $(X,\Sigma ',\nu$) is a complete measure space.

  A metric outer measure is an outer measure ν defined on the subsets of a given metric space (X, d) such that ν(A ∪ B) = ν(A) +ν(B) for every pair of nonempty subsets A, B ⊂ X with positive set-set distance inf _{a ∈ A, b ∈ B} d(a, b). An example is Hausdorff q-measure; cf. Hausdorff dimension.

• Length of metric space

  The Fremlin length of a metric space (X, d) is its Hausdorff 1-measure H ¹(X).

  The Hejcman length  lng(M) of a subset M ⊂ X of a metric space (X, d) is sup{lng(M′): M′ ⊂ M, | M′ |  < ∞}. Here lng(∅) = 0 and, for a finite subset M′ ⊂ X, lng(M′) = min∑ _i = 1 ⁿ d(x _i−1, x _i ) over all sequences $x_{0},\ldots,x_{n}$ such that $\{x_{i}: i = 0,1,\ldots,n\} = M'$.

  The Schechtman length of a finite metric space (X, d) is $\inf \sqrt{\sum _{i=1 }^{n }a_{i }^{2}}$ over all sequences $a_{1},\ldots,a_{n}$ of positive numbers such that there exists a sequence $X_{0},\ldots,X_{n}$ of partitions of X with following properties:

  • 1. X ₀ = { X} and X _n  = {{ x}: x ∈ X};

  • 2. X _i refines X _i−1 for $i = 1,\ldots,n$;

  • 3. For $i = 1,\ldots,n$ and B, C ⊂ A ∈ X _i−1 with B, C ∈ X _i , there exists a one-to-one map f from B onto C such that d(x, f(x)) ≤ a _i for all x ∈ B.

• Volume of finite metric space

  Given a metric space (X, d) with | X |  = k < ∞, its volume (Feige, 2000) is the maximal (k − 1)-dimensional volume of the simplex with vertices { f(x): x ∈ X} over all metric mappings $f: (X,d) \rightarrow (\mathbb{R}^{k-1},l_{2})$. The volume coincides with the metric for k = 2. It is monotonically increasing and continuous in the metric d.

• Rank of metric space

  The Minkowski rank of a metric space (X, d) is the maximal dimension of a normed vector space (V, | | . | | ) such that there is an isometry (V, | | . | | ) → (X, d).

  The Euclidean rank of a metric space (X, d) is the maximal dimension of a flat in it, that is of a Euclidean space $\mathbb{E}^{n}$ such that there is an isometric embedding $\mathbb{E}^{n} \rightarrow (X,d)$.

  The quasi-Euclidean rank of a metric space (X, d) is the maximal dimension of a quasi-flat in it, i.e., of an Euclidean space $\mathbb{E}^{n}$ admitting a quasi-isometry $\mathbb{E}^{n} \rightarrow (X,d)$. Every Gromov hyperbolic metric space has this rank 1.

• Roundness of metric space

  The roundness of a metric space (X, d) is the supremum of all q such that

  $$\displaystyle{d(x_{1},x_{2})^{q} + d(y_{ 1},y_{2})^{q} \leq d(x_{ 1},y_{1})^{q} + d(x_{ 1},y_{2})^{q} + d(x_{ 2},y_{1})^{q} + d(x_{ 2},y_{2})^{q}}$$

  for any four points x ₁, x ₂, y ₁, y ₂ ∈ X.

  Every metric space has roundness ≥ 1; it is ≤ 2 if the space has approximate midpoints. The roundness of L _p -space is p if 1 ≤ p ≤ 2.

  The generalized roundness of a metric space (X, d) is (Enflo, 1969) the supremum of all q such that, for any 2k ≥ 4 points x _i , y _i  ∈ X with 1 ≤ i ≤ k,

  $$\displaystyle{\sum _{1\leq i<j\leq k}d^{q}(x_{ i},x_{j}) + d^{q}(y_{ i},y_{j}) \leq \sum _{1\leq i,j\leq k}d^{q}(x_{ i},y_{j}).}$$

  Lennard–Tonge–Weston, 1997, have shown that the generalized roundness is the supremum of q such that d is of q-negative type, i.e., d ^q is of negative type.

  Every CAT(0) space (Chap. 6) has roundness 2, but some of them have generalized roundness 0 (Lafont–Prassidis, 2006).

• Type of metric space

  The Enflo type of a metric space (X, d) is p if there exists a constant 1 ≤ C < ∞ such that, for every $n \in \mathbb{N}$ and every function f: { −1, 1}ⁿ → X, $\sum _{\epsilon \in \{-1,1\}^{n}}d^{p}(\,f(\epsilon ),f(-\epsilon ))$ is at most

  $C^{p}\sum _{j=1}^{n}\sum _{\epsilon \in \{-1,1\}^{n}}d^{p}(\,f(\epsilon _{1},\ldots,\,\epsilon _{j\,-\,1},\epsilon _{j},\epsilon _{j\,+\,1},\ldots,\epsilon _{n}),\,f(\epsilon _{1},\ldots,\,\epsilon _{j-1},\,-\epsilon _{j},\epsilon _{j+1},\ldots,\epsilon _{n}))$.

  A Banach space (V, | | . | | ) of Enflo type p has Rademacher type p, i.e., for every $x_{1},\ldots,x_{n} \in V$, it holds

  $$\displaystyle{\sum _{\epsilon \in \{-1,1\}^{n}}\vert \vert \sum _{j=1}^{n}\epsilon _{ j}x_{j}\vert \vert ^{p} \leq C^{p}\sum _{ j=1}^{n}\vert \vert x_{ j}\vert \vert ^{p}.}$$

  Given a metric space (X, d), a symmetric Markov chain on X is a Markov chain {Z _l } _l = 0 ^∞ on a state space $\{x_{1},\ldots,x_{m}\} \subset X$ with a symmetrical transition m × m matrix ((a _ij )), such that P(Z _l+1 = x _j : Z _l  = x _i ) = a _ij and $P(Z_{0} = x_{i}) = \frac{1} {m}$ for all integers 1 ≤ i, j ≤ m and l ≥ 0. A metric space (X, d) has Markov type p (Ball, 1992) if sup _T M _p (X, T) < ∞ where M _p (X, T) is the smallest constant C > 0 such that the inequality

  $$\displaystyle{\mathbb{E}d^{p}(Z_{ T},Z_{0}) \leq TC^{p}\mathbb{E}d^{p}(Z_{ 1},Z_{0})}$$

  holds for every symmetric Markov chain {Z _l } _l = 0 ^∞ on X holds, in terms of expected value (mean) $\mathbb{E}[X] =\sum _{x}xp(x)$ of the discrete random variable X.

  A metric space of Markov type p has Enflo type p.

• Strength of metric space

  Given a finite metric space (X, d) with s different nonzero values of d _ij  = d(i, j), its strength is the largest number t such that, for any integers p, q ≥ 0 with p + q ≤ t, there is a polynomial f _pq (s) of degree at most min{p, q} such that ((d _ij ^2p))((d _ij ^2q)) = (( f _pq (d _ij ²))).

• Rendez-vous number

  Given a metric space (X, d), its rendez-vous number (or Gross number, magic number) is a positive real number r(X, d) (if it exists) defined by the property that for each integer n and all (not necessarily distinct) $x_{1},\ldots,x_{n} \in X$ there exists a point x ∈ X such that

  $$\displaystyle{r(X,d) = \frac{1} {n}\sum _{i=1}^{n}d(x_{ i},x).}$$

  If the number r(X, d) exists, then it is said that (X, d) has the average distance property . Every compact connected metric space has this property. The unit ball {x ∈ V:  | | x | | ≤ 1} of a Banach space (V, | | . | | ) has the rendez-vous number 1.

• Wiener-like distance indices

  Given a finite subset M of a metric space (X, d) and a parameter q, the Wiener polynomial of M (as defined by Hosoya, 1988, for the graphic metric d _path) is

  $$\displaystyle{W(M;q) = \frac{1} {2}\sum _{x,y\in M:\,x\neq y}q^{d(x,y)}.}$$

  It is a generating function for the distance distribution (Chap. 16) of M, i.e., the coefficient of q ⁱ in W(M; q) is the number | {{x, y} ∈ M × M: d(x, y) = i} | .

  In the main case when M is the vertex-set V of a connected graph G = (V, E) and d is the path metric of G, the number $W(M;1) = \frac{1} {2}\sum _{x,y\in M}d(x,y)$ is called the Wiener index of G. This notion is originated (Wiener, 1947) and applied, together with its many analogs, in Chemistry; cf. chemical distance in Chap. 24.

  The hyper-Wiener index is ∑ _{x, y ∈ M} (d(x, y) + d(x, y)²). The reverse-Wiener index is $\frac{1} {2}\sum _{x,y\in M}(D - d(x,y))$, where D is the diameter of M. The complementary reciprocal Wiener index is $\frac{1} {2}\sum _{x,y\in M}(1 + D - d(x,y))^{-1}$. The Harary index is ∑ _{x, y ∈ M} (d(x, y))⁻¹. The Szeged index and the vertex PI index are ∑ _e ∈ E n _x (e)n _y (e) and ∑ _e ∈ E (n _x (e) + n _y (e)), where e = (xy) and n _x (e)= | {z ∈ V: d(x, z) < d(y, z)} | .

  Two studied edge-Wiener indices of G are the Wiener index of its line graph and ∑ _{(xy), (x′y′) ∈ E} max{d(x, x′), d(x, y′), d(y, x′), d(y, y′)}.

  The Gutman–Schultz index, degree distance (Dobrynin–Kochetova, 1994), reciprocal degree distance and terminal Wiener index are:

  $$\displaystyle{\sum _{x,y\in M}r_{x}r_{y}d(x,y),\sum _{x,y\in M}d(x,y)(r_{x}+r_{y}),\sum _{x,y\in M} \frac{1} {d(x,y)}(r_{x}+r_{y}),\sum _{x,y\in \{z\in M:r_{z}=1\}}d(x,y),}$$

  where r _z is the degree of the vertex z ∈ M. The eccentric distance sum (Gupta et al., 2002) is ∑ _y ∈ M (max{d(x, y): x ∈ M}d _y ), where d _y is ∑ _x ∈ M d(x, y). The Balaban index is $\frac{\vert E\vert } {c+1}\sum _{(yz)\in E}(\sqrt{d_{y } d_{z}})^{-1}$, where c is the number of primitive cycles. The multiplicative Wiener index is (Das–Gutman, 2016) ∏ _{x, y ∈ M, x ≠ y} d(x, y).

  Given a partition $P =\{ V _{1},\ldots,V _{k}\}$ of the vertex-set V, set f _P (x) = i for x ∈ V _i . The colored distance (Dankelman et al., 2001) and the partition distance (Klavžar, 2016) of G are $\sum _{f_{P}(x)\neq f_{P}(y)}d(x,y)$ and $\sum _{f_{P}(x)=f_{P}(y)}d(x,y)$, respectively.

  Above indices are called (corresponding) Kirchhoff indices if d the resistance metric (Chap. 15) of G.

  The average distance of M is the number $\frac{1} {\vert M\vert (\vert M\vert -1)}\sum _{x,y\in M}d(x,y)$. In general, for a quasi-metric space (X, d), the numbers ∑ _{x, y ∈ M} d(x, y) and $\frac{1} {\vert M\vert (\vert M\vert -1)}\sum _{x,y\in M,x\neq y} \frac{1} {d(x,y)}$ are called, respectively, the transmission and global efficiency of M.

• Distance polynomial

  Given an ordered finite subset M of a metric space (X, d), let D be the distance matrix of M. The distance polynomial of M is the characteristic polynomial of D, i.e., the determinant det(D −λ I).

  Usually, D is the distance matrix of the path metric of a graph. Sometimes, the distance polynomial is defined as det(λ I − D) or (−1)ⁿ det(D −λ I).

  The roots of the distance polynomial constitute the distance spectrum (or D-spectrum of D-eigenvalues) of M. Let ρ _max and ρ _min be the largest and the smallest roots; then ρ _max and ρ _max −ρ _min are called (distance spectral) radius and spread of M. The distance degree of x ∈ M is ∑ _y ∈ M d(x, y). The distance energy of M is the sum of the absolute values of its D-eigenvalues. It is 2ρ _max if (as, for example, for the path metric of a tree) exactly one D-eigenvalue is positive.

• s-energy

  Given a finite subset M of a metric space (X, d) and a number s > 0, the s-energy and 0-energy of M are, respectively, the numbers

  $$\displaystyle{\sum _{x,y\in M,x\neq y} \frac{1} {d^{s}(x,y)}\,\,\,\mbox{ and}\,\,\,\sum _{x,y\in M,x\neq y}\log \frac{1} {d(x,y)} = -\log \prod _{x,y\in M,x\neq y}d(x,y).}$$

  The (unnormalized) s-moment of M is the number ∑ _{x, y ∈ M} d ^s(x, y).

  The discrete Riesz s-energy is the s-energy for Euclidean distance d. In general, let μ be a finite Borel probability measure on (X, d). Then $U_{s}^{\mu }(x) =\int \frac{\mu (dy)} {d(x,y)^{s}}$ is the (abstract) s-potential at a point x ∈ X. The Newton gravitational potential is the case $(X,d) = (\mathbb{R}^{3},\vert x - y\vert )$, s = 1, for the mass distribution μ.

  The s-energy of μ is $E_{s}^{\mu } =\int U_{s}^{\mu }(x)\mu (dx) =\int \int \frac{\mu (dx)\mu (dy)} {d(x,y)^{s}}$, and the s-capacity of (X, d) is (inf _μ E _s ^μ)⁻¹. Cf. the metric capacity.

• Fréchet mean

  Given a metric space (X, d) and a number s > 0, the Fréchet function is $F_{s}(x) = \mathbb{E}[d^{s}(x,y)]$. For a finite subset M of X, this expected value is the mean F _s (x) = ∑ _y ∈ M w(y)d ^s(x, y), where w(y) is a weight function on M.

  The points, minimizing F ₁(x) and F ₂(x), are called the Fréchet median (or weighted geometric median) and Fréchet mean (or Karcher mean), respectively.

  If $(X,d) = (\mathbb{R}^{n},\vert \vert x - y\vert \vert _{2})$ and the weights are equal, these points are called the geometric median (or Fermat–Weber point, 1-median) and the centroid (or geometric center, barycenter), respectively.

  The k-median and k-mean of M are the k-sets C minimizing, respectively, the sums ∑ _y ∈ M min _c ∈ C d(y, c) = ∑ _y ∈ M d(y, C) and ∑ _y ∈ M d ²(y, C).

  Let (X, d) be the metric space $(\mathbb{R}_{>0},\vert \,f(x) - f(y)\vert )$, where $f: \mathbb{R}_{>0} \rightarrow \mathbb{R}$ is a given injective and continuous function. Then the Fréchet mean of $M \subset \mathbb{R}_{>0}$ is the f-mean (or Kolmogorov mean, quasi-arithmetic mean) $f^{-1}(\frac{\sum _{x\in M}f(x)} {\vert M\vert } )$. It is the arithmetic, geometric, harmonic, and power mean if $f = x,\log (x), \frac{1} {x}$, and f = x ^p (for a given p ≠ 0), respectively. The cases p → +∞, p → −∞ correspond to maximum and minimum, while p = 2, = 1, → 0, → −1 correspond to the quadratic, arithmetic, geometric and harmonic mean.

  Given a completely monotonic (i.e., (−1)^k f ^(k) ≥ 0 for any k) function $f \in \mathbb{C}^{\infty }$, the f-potential energy of a finite subset M of (X, d) is ∑ _{x, y ∈ M, x ≠ y} f(d ²(x, y)). The set M is called (Cohn–Kumar, 2007) universally optimal if it minimizes, among sets M′ ⊂ X with | M′ |  =  | M | , the f-potential energy for any such f. Among universally optimal subsets of $(\mathbb{S}^{n-1},\vert \vert x - y\vert \vert _{2})$, there are the vertex-sets of a polygon, simplex, cross-polytope, icosahedron, 600-cell, E ₈ root system.

• Distance-weighted mean

  In Statistics, the distance-weighted mean between given data points $x_{1},\ldots,x_{n}$ is defined (Dodonov–Dodonova, 2011) by

  $$\displaystyle{\frac{\sum _{1\leq i\leq n}w_{i}x_{i}} {\sum _{1\leq i\leq n}w_{i}} \,\,\mbox{ with }\,\,w_{i} = \frac{n - 1} {\sum _{1\leq j\leq n}\vert x_{i} - x_{j}\vert }.}$$

  The case w _i  = 1 for all i corresponds to the arithmetic mean.

• Inverse distance weighting

  In Numerical Analysis, multivariate (or spatial) interpolation is interpolation on functions of more than one variable. Inverse distance weighting is a method (Shepard, 1968) for multivariate interpolation. Let $x_{1},\ldots,x_{n}$ be interpolating points (i.e., samples u _i  = u(x _i ) are known), x be an interpolated (unknown) point and d(x, x _i ) be a given distance. A general form of interpolated value u(x) is

  $$\displaystyle{u(x) = \frac{\sum _{1\leq i\leq n}w_{i}(x)u_{i}} {\sum _{1\leq i\leq n}w_{i}(x)},\mbox{ with }w_{i}(x) = \frac{1} {(d(x,x_{i}))^{p}},}$$

  where p > 0 (usually p = 2) is a fixed power parameter.

• Transfinite diameter

  The n-th diameter D _n (M) and the n-th Chebyshev constant C _n (M) of a set M ⊆ X in a metric space (X, d) are defined (Fekete, 1923, for the complex plane $\mathbb{C}$) as

  $$\displaystyle{D_{n}(M) =\sup _{x_{1},\ldots,x_{n}\in M}\prod _{i\neq j}d(x_{i},x_{j})^{ \frac{1} {n(n-1)} }\,\,\mbox{ and}\,\,C_{n}(M) =\inf _{x\in X}\sup _{x_{ 1},\ldots,x_{n}\in M}\prod _{j=1}^{n}d(x,x_{ j})^{ \frac{1} {n} }.}$$

  The number logD _n (M) (the supremum of the average distance) is called the n-extent of M. The numbers D _n (M), C _n (M) come from the geometric mean averaging; they also come as the limit case s → 0 of the s-moment ∑ _{i ≠ j} d(x _i , x _j )^s averaging.

  The transfinite diameter (or ∞-th diameter) and the ∞-th Chebyshev constant C _∞ (M) of M are defined as

  $$\displaystyle{D_{\infty }(M) =\lim _{n\rightarrow \infty }D_{n}(M)\,\,\mbox{ and}\,\,C_{\infty }(M) =\lim _{n\rightarrow \infty }C_{n}(M);}$$

  these limits existing since {D _n (M)} and {C _n (M)} are nonincreasing sequences of nonnegative real numbers. Define D _∞ (∅) = 0.

  The transfinite diameter of a compact subset of $\mathbb{C}$ is its conformal radius at infinity (cf. Chap. 6); for a segment in $\mathbb{C}$, it is $\frac{1} {4}$ of its length.

• Metric diameter

  The metric diameter (or diameter, width) diam(M) of a set M ⊆ X in a metric space (X, d) is defined by

  $$\displaystyle{\sup _{x,y\in M}d(x,y).}$$

  The diameter graph of M has, as vertices, all points x ∈ M with d(x, y) = diam(M) for some y ∈ M; it has, as edges, all pairs of its vertices at distance diam(M) in (X, d). (X, d) is called a diametrical metric space if any x ∈ X has the antipode, i.e., a unique x′ ∈ X such that the closed metric interval I(x, x′) is X.

  The furthest neighbor digraph of M is a directed graph on M, where xy is an arc (called a furthest neighbor pair) whenever y is at maximal distance from x.

  In a metric space endowed with a measure, one says that the isodiametric inequality holds if the metric balls maximize the measure among all sets with given diameter. It holds for the volume in Euclidean space but not, for example, for the Heisenberg metric on the Heisenberg group (Chap. 10).

  The k-ameter (Grove–Markvorsen, 1992) is $\sup _{K\subseteq X:\,\vert K\vert =k}\frac{1} {2}\sum _{x,y\in K}d(x,y)$, and the k-diameter (Chung–Delorme–Sole, 1999) is sup _{K ⊆ X:  | K | = k} inf _{x, y ∈ K: x ≠ y} d(x, y).

  Given a property P ⊆ X × X of a pair (K, K′) of subsets of a finite metric space (X, d), the conditional diameter (called P-diameter in Balbuena et al., 1996) is max_{(K, K′) ∈ P} min_{(x, y) ∈ K×K′} d(x, y). It is diam(X, d) if P = { (K, K′) ∈ X × X:  | K |  =  | K′ |  = 1}. When (X, d) models an interconnection network, the P-diameter corresponds to the maximum delay of the messages interchanged between any pair of clusters of nodes, K and K′, satisfying a given property P of interest.

• Metric spread

  A subset M of a metric space (X, d) is called Delone set (or separated ε -net, (A,a)-Delone set) if it is bounded (with a finite diameter A = sup _{x, y ∈ M} d(x, y)) and metrically discrete (with a separation a = inf _{x, y ∈ M, x ≠ y} d(x, y) > 0).

  The metric spread (or distance ratio, normalized diameter) of M is the ratio $\frac{A}{a}$.

  The aspect ratio (or axial ratio) of a shape is the ratio of its longer and shorter dimensions, say, the length and diameter of a rod, major and minor axes of a torus or width and height of a rectangle (image, display, pixel, etc.).

  For a mesh M with separation a and covering radius (or mesh norm) c = sup _y ∈ X inf _x ∈ M d(x, y), the mesh ratio is $\frac{c} {a}$.

  In Physics, the aspect ratio is the ratio of height-to-length scale characteristics. Cf. the wing’s aspect ratio among aircraft distances in Chap. 29.

  Dynamic range DNR is the ratio between the largest and smallest possible values of a quantity, such as in sound or light signals; cf. SNR distance in Chap. 21.

• Eccentricity

  Given a bounded metric space (X, d), the eccentricity (or Koenig number) of a point x ∈ X is the number e(x) = max _y ∈ X d(x, y).

  The numbers D = max _x ∈ X e(x) and r = min _x ∈ X e(x) are called the diameter and the radius of (X, d), respectively. The point z ∈ X is called central if e(z) = r, peripheral if e(z) = D, and pseudo-peripheral if for each point x with d(z, x) = e(z) it holds that e(z) = e(x). For finite | X | , the average eccentricity is $\frac{1} {\vert X\vert }\sum _{x\in X}e(x)$, and the contour of (X, d) is the set of points x ∈ X such that no neighbor (closest point) of x has an eccentricity greater than x.

  The eccentric digraph (Buckley, 2001) of (X, d) has, as vertices, all points x ∈ X and, as arcs, all ordered pairs (x, y) of points with d(x, y) = e(y). The eccentric graph (Akyiama–Ando–Avis, 1976) of (X, d) has, as vertices, all points x ∈ X and, as edges, all pairs (x, y) of points at distance min{e(x), e(y)}.

  The super-eccentric graph (Iqbalunnisa–Janairaman–Srinivasan, 1989) of (X, d) has, as vertices, all points x ∈ X and, as edges, all pairs (x, y) of points at distance no less than the radius of (X, d). The radial graph (Kathiresan–Marimuthu, 2009) of (X, d) has, as vertices, all points x ∈ X and, as edges, all pairs (x, y) of points at distance equal to the radius of (X, d).

  The sets {x ∈ X: e(x) ≤ e(z) for any z ∈ X}, {x ∈ X: e(x) ≥ e(z) for any z ∈ X} and {x ∈ X: ∑ _y ∈ X d(x, y) ≤ ∑ _y ∈ X d(z, y) for any z ∈ X} are called, respectively, the metric center (or eccentricity center, center), metric antimedian (or periphery) and the metric median (or distance center) of (X, d).

• Radii of metric space

  Given a bounded metric space (X, d) and a set M ⊆ X of diameter D, its metric radius (or radius ) Mr, covering radius (or directed Hausdorff distance from X to M) Cr and remoteness (or Chebyshev radius ) Re are the numbers inf _x ∈ M sup _y ∈ M d(x, y), sup _x ∈ X inf _y ∈ M d(x, y) and inf _x ∈ X sup _y ∈ M d(x, y), respectively. It holds that $\frac{D} {2} \leq Re \leq Mr \leq D$ with $Mr = \frac{D} {2}$ in any injective metric space. Somemimes, $\frac{D} {2}$ is called the radius.

  For m > 0, a minimax distance design of size m is an m-subset of X having smallest covering radius. This radius is called the m-point mesh norm of (X, d).

  The packing radius Pr of M is the number sup{r: inf _{x, y ∈ M, x ≠ y} d(x, y) > 2r}. For m > 0, a maximum distance design of size m is an m-subset of X having largest packing radius. This radius is the m-point best packing distance on (X, d).

• ε -net

  Given a metric space (X, d), a subset M ⊂ X, and a number ε > 0, the ε -neighborhood of M is the set M ^ε = ∪ _x ∈ M B(x, ε).

  The set M is called an ε -net (or ε-covering, ε-approximation) of (X, d) if M ^ε = X, i.e., the covering radius of M is at most ε.

  Let C _ε denote the ε-covering number, i.e., the smallest size of an ε-net in (X, d). The number lg₂ C _ε is called (Kolmogorov–Tikhomirov, 1959) the metric entropy (or ε-entropy) of (X, d). It holds $P_{\epsilon } \leq C_{\epsilon } \leq P_{ \frac{\epsilon }{ 2} }$, where P _ε denote the ε-packing number of (X, d), i.e., $\sup \{\vert M\vert: M \subset X,\overline{B}(x,\epsilon ) \cap \overline{B}(y,\epsilon ) =\emptyset \mbox{ for any }x,y \in M,x\neq y\}$. The number lg₂ P _ε is called the metric capacity (or ε-capacity) of (X, d).

• Steiner ratio

  Given a metric space (X, d) and a finite subset V ⊂ X, let G = (V, E) be the complete weighted graph on V with edge-weights d(x, y) for all x, y ∈ V.

  Given a tree T, its weight is the sum d(T) of its edge-weights. A spanning tree of V is a subset of | V | − 1 edges forming a tree on V. Let MSpT _V be a minimum spanning tree of V, i.e., a spanning tree with the minimal weight d(MSpT _V ).

  A Steiner tree of V is a tree on Y, V ⊂ Y ⊂ X, connecting vertices from V; elements of Y∖V are called Steiner points. Let StMT _V be a minimum Steiner tree of V, i.e., a Steiner tree with the minimal weight d(StMT _V ) = inf _{Y ⊂ X : V ⊂ Y} d(MSpT _Y ). This weight is called the Steiner diversity of V; cf. diversity in Chap. 3 It is the Steiner distance of set V (Chap. 15) if (X, d) is graphic metric space.

  The Steiner ratio St(X, d) of the metric space (X, d) is defined by

  $$\displaystyle{\inf _{V \subset X} \frac{d(StMT_{V })} {d(MSpT_{V })}.}$$

  Cf. arc routing problems in Chap. 15

• Chromatic numbers of metric space

  Given a metric space (X, d) and a set D of positive real numbers, the D-chromatic number of (X, d) is the standard chromatic number of its D-distance graph, i.e., the graph (X, E) with the vertex-set X and the edge-set E = { xy: d(x, y) ∈ D} (Chap. 15). Usually, (X, d) is an l _p -space and D = { 1} (Benda–Perles chromatic number ) or D = [1 −ε, 1 +ε].

  For a metric space (X, d), the polychromatic number is the minimum number of colors needed to color all the points x ∈ X so that, for each color class C _i , there is a distance d _i such that no two points of C _i are at distance d _i .

  For a metric space (X, d), the packing chromatic number is the minimum number of colors needed to color all the points x ∈ X so that, for each color class C _i , no two distinct points of C _i are at distance at most i.

  For any integer t > 0, the t-distance chromatic number of a metric space (X, d) is the minimum number of colors needed to color all the points x ∈ X so that any two points whose distance is ≤ t have distinct colors. Cf. k-distance chromatic number in Chap. 15

  For any integer t > 0, the t-th Babai number of a metric space (X, d) is the minimum number of colors needed to color all the points in X so that, for any set D of positive distances with | D | ≤ t, any two points x, y ∈ X with d(x, y) ∈ D have distinct colors.

• Congruence order of metric space

  A metric space (X, d) has congruence order n if every finite metric space which is not isometrically embeddable in (X, d) has a subspace with at most n points which is not isometrically embeddable in (X, d). For example, the congruence order of l ₂ ⁿ is n + 3 (Menger, 1928); it is 4 for the path metric of a tree.

4 Main Mappings of Metric Spaces

• Distance function

  In Topology, the term distance function is often used for distance. But, in general, a distance function (or ray function) is a continuous function on a metric space (X, d) (usually, on a Euclidean space $\mathbb{E}^{n}$) $f: X \rightarrow \mathbb{R}_{\geq 0}$ which is homogeneous, i.e., f(tx) = tf(x) for all t ≥ 0 and all x ∈ X.

  Such function f is called positive if f(x) > 0 for all x ≠ 0, symmetric if f(x) = f(−x), convex if f(tx + (1 − t)y) ≤ tf(x) + (1 − t)f(y) for any 0 < t < 1 and x ≠ y, and strictly convex if this inequality is strict.

  If $X = \mathbb{E}^{n}$, the set $S_{f} =\{ x \in \mathbb{R}^{n}: f(x) < 1\}$ is star body, i.e., x ∈ S _f implies [0, x] ⊂ S _f . Any star body S corresponds to a unique distance function $g(x) =\inf _{tx\in S,t>0}\frac{1} {t}$, and S = S _g . The star body is bounded if f is positive, symmetric about the origin if f is symmetric, convex if f is convex, and strictly convex (i.e., the boundary ∂ B does not contain a segment) if f is strictly convex.

  For a quadratic distance function of the form f _A  = xAx ^T, where A is a real matrix and $x \in \mathbb{R}^{n}$, the matrix A is positive-definite (i.e., the Gram matrix VV ^T = ((〈v _i , v _j 〉)) of n linearly independent vectors $v_{i} = (v_{i1},\ldots,v_{in})$) if and only if f _A is symmetric and strictly convex function. The homogeneous minimum of f _A is

  $$\displaystyle{\min (\,f_{A}) =\inf _{x\in \mathbb{Z}^{n}\setminus \{0\}}f_{A}(x) =\inf _{x\in L\setminus \{0\}}\sum _{1\leq i\leq n}x_{i}^{2},}$$

  where $L =\{\sum x_{i}v_{i}: x_{i} \in \mathbb{Z}\}$ is a lattice, i.e., a discrete subgroup of $\mathbb{R}^{n}$ spanning it. The Hermite constant γ _n , a central notion in Geometry of Numbers, is the supremum, over all positive-definite (n × n)-matrices, of $\min (\,f_{A})\det (A)^{ \frac{1} {n} }$. It is known only for 2 ≤ n ≤ 8 and n = 24; cf. systole of metric space.

• Convex distance function

  Given a compact convex region $B \subset \mathbb{R}^{n}$ containing the origin O in its interior, the convex distance function (or Minkowski distance function , Minkowski seminorm, gauge) is the function | | P | |  _B whose value at a point $P \in \mathbb{R}^{n}$ is the distance ratio $\frac{OP} {OQ}$, where Q ∈ B is the furthest from O point on the ray OP.

  Then d _B (x, y) =  | | x − y | |  _B is the quasi-metric on $\mathbb{R}^{n}$ defined, for x ≠ y, by

  $$\displaystyle{\inf \{\alpha > 0: y - x \in \alpha B\},}$$

  and $B =\{ x \in \mathbb{R}^{n}: d_{B}(0,x) \leq 1\}$ with equality only for x ∈ ∂ B.

  The function | | P | |  _B is called a polyhedral distance function if B is a n-polytope, simplicial distance function if it is a n-simplex, and so on.

  If B is centrally-symmetric with respect to the origin, then d _B is a Minkowskian metric (Chap. 6) whose unit ball is B. This is the l ₁-metric if B is the n-cross-polytope and the l _∞ -metric if B is the n-cube.

• Funk distance

  Let B be an nonempty open convex subset of $\mathbb{R}^{n}$. For any x, y ∈ B, denote by R(x, y) the ray from x through y. The Funk distance (Funk, 1929) on B is the quasi-semimetric defined, for any x, y ∈ B, as 0 if the boundary ∂(B) and R(x, y) are disjoint, and, otherwise, i.e., if R(x, y) ∩ ∂ B = { z}, by

  $$\displaystyle{\ln \frac{\vert \vert x - z\vert \vert _{2}} {\vert \vert y - z\vert \vert _{2}}.}$$

  The Hilbert projective metric in Chap. 6 is a symmetrization of this distance.

• Metric projection

  Given a metric space (X, d) and a subset M ⊂ X, an element u ₀ ∈ M is called an element of best approximation (or nearest point ) to a given element x ∈ X if d(x, u ₀) = inf _u ∈ M d(x, u), i.e., if d(x, u ₀) is the point-set distance d(x, M).

  A metric projection (or operator of best approximation, nearest point map) is a multivalued mapping associating to each element x ∈ X the set of elements of best approximation from the set M (cf. distance map ).

  A Chebyshev set in a metric space (X, d) is a subset M ⊂ X containing a unique element of best approximation for every x ∈ X.

  A subset M ⊂ X is called a semi-Chebyshev set if the number of such elements is at most one, and a proximinal set if this number is at least one.

  While the Chebyshev radius (or remoteness; cf. radii of metric space ) of the set M is inf _x ∈ X sup _y ∈ M d(x, y), a Chebyshev center of M is an element x ₀ ∈ X realizing this infimum. Sometimes (say, for a finite graphic metric space), $\frac{1} {\vert M\vert }\inf _{x\in X}\sum _{y\in M}d(x,y)$ and $\frac{1} {\vert M\vert }\sup _{x\in X}\sum _{y\in M}d(x,y)$ are called proximity and remoteness of M.

• Distance map

  Given a metric space (X, d) and a subset M ⊂ X, the distance map is a function $f_{M}: X \rightarrow \mathbb{R}_{\geq 0}$, where f _M (x) = inf _u ∈ M d(x, u) is the point-set distance d(x, M) (cf. metric projection ).

  If the boundary B(M) of the set M is defined, then the signed distance function g _M is defined by g _M (x) = −inf _{u ∈ B(M)} d(x, u) for x ∈ M, and g _M (x) = inf _{u ∈ B(M)} d(x, u), otherwise. If M is a (closed orientable) n-manifold (Chap. 2), then g _M is the solution of the eikonal equation | ∇g |  = 1 for its gradient ∇.

  If $X = \mathbb{R}^{n}$ and, for every x ∈ X, there is unique element u(x) with d(x, M) = d(x, u(x)) (i.e., M is a Chebyshev set ), then | | x − u(x) | | is called a vector distance function .

  Distance maps are used in Robot Motion (M being the set of obstacle points) and, especially, in Image Processing (M being the set of all or only boundary pixels of the image). For $X = \mathbb{R}^{2}$, the graph {(x, f _M (x)): x ∈ X} of d(x, M) is called the Voronoi surface of M.

• Isometry

  Given metric spaces (X, d _X ) and (Y, d _Y ), a function f: X → Y is called an isometric embedding of X into Y if it is injective and the equality d _Y ( f(x), f(y)) = d _X (x, y) holds for all x, y ∈ X.

  An isometry (or congruence mapping) is a bijective isometric embedding. Two metric spaces are called isometric (or isometrically isomorphic) if there exists an isometry between them.

  A property of metric spaces which is invariant with respect to isometries (completeness, boundedness, etc.) is called a metric property (or metric invariant).

  A path isometry (or arcwise isometry) is a mapping from X into Y (not necessarily bijective) preserving lengths of curves.

• Rigid motion of metric space

  A rigid motion (or, simply, motion ) of a metric space (X, d) is an isometry of (X, d) onto itself.

  For a motion f, the displacement function d _f (x) is d(x, f(x)). The motion f is called semisimple if inf _x ∈ X d _f (x) = d(x ₀, f(x ₀)) for some x ₀ ∈ X, and parabolic, otherwise. A semisimple motion is called elliptic if inf _x ∈ X d _f (x) = 0, and axial (or hyperbolic), otherwise. A motion is called a Clifford translation if the displacement function d _f (x) is a constant for all x ∈ X.

• Symmetric metric space

  A metric space (X, d) is called symmetric if, for any point p ∈ X, there exists a symmetry relative to that point, i.e., a motion f _p of this metric space such that f _p ( f _p (x)) = x for all x ∈ X, and p is an isolated fixed point of f _p .

• Homogeneous metric space

  A metric space is called homogeneous (or point-homogeneous) if, for any two points of it, there exists a motion mapping one of the points to the other.

  In general, a homogeneous space is a set together with a given transitive group of symmetries. Moss, 1992, defined similar distance-homogeneous distanced graph.

  A metric space is called ultrahomogeneous space (or highly transitive) if any isometry between two of its finite subspaces extends to the whole space.

  A metric space (X, d) is called (Grünbaum–Kelly) a metrically homogeneous metric space if {d(x, z): z ∈ X} = { d(y, z): z ∈ X} for any x, y ∈ X.

• Flat space

  A flat space is any metric space with local isometry to some $\mathbb{E}^{n}$, i.e., each point has a neighborhood isometric to an open set in $\mathbb{E}^{n}$. A space is locally Euclidean if every point has a neighborhood homeomorphic to an open subset in $\mathbb{E}^{n}$.

• Dilation of metric space

  Given a metric space (X, d), its dilation (or r-dilation ) is a mapping f: X → X with d( f(x), f(y)) = rd(x, y) for some r > 0 and any x ∈ X.

• Wobbling of metric space

  Given a metric space (X, d), its wobbling (or r-wobbling ) is a mapping f: X → X with d(x, f(x)) < r for some r > 0 and any x ∈ X.

• Paradoxical metric space

  Given a metric space (X, d) and an equivalence relation on the subsets of X, the space (X, d) is called paradoxical if X can be decomposed into two disjoint sets M ₁, M ₂ so that M ₁, M ₂ and X are pairwise equivalent.

  Deuber, Simonovitz and Sós, 1995, introduced this idea for wobbling equivalent subsets M ₁, M ₂ ⊂ X, i.e., there is a bijective r-wobbling f: M ₁ → M ₂. For example, $(\mathbb{R}^{2},l_{2})$ is paradoxical for wobbling but not for isometry equivalence.

• Metric cone

  A pointed metric space (X, d, x ₀) is called a metric cone, if it is isometric to (λ X, d, x ₀) for all λ > 0. A metric cone structure on (X, d, x ₀) is a (pointwise) continuous family f _t ($t \in \mathbb{R}_{>0}$) of dilations of X, leaving the point x ₀ invariant, such that d( f _t (x), f _t (y)) = td(x, y) for all x, y and f _t ∘ f _s  = f _ts . A Banach space has such a structure for the dilations f _t (x) = tx ($t \in \mathbb{R}_{>0}$). The Euclidean cone over a metric space (cf. cone over metric space in Chap. 9) is another example.

  The tangent metric cone over a metric space (X, d) at a point x ₀ is (for all dilations tX = (X, td)) the closure of ∪ _t > 0 tX, i.e., of lim _{t → ∞} tX taken in the pointed Gromov–Hausdorff topology (cf. Gromov–Hausdorff metric ).

  The asymptotic metric cone over (X, d) is its tangent metric cone “at infinity”, i.e., ∩ _t > 0 tX = lim _t → 0 tX. Cf. boundary of metric space in Chap. 6.

  The term metric cone was also used by Bronshtein, 1998, for a convex cone C equipped with a complete metric compatible with its operations of addition (continuous on C × C) and multiplication (continuous on $C \times \mathbb{R}_{\geq 0}$). by all λ ≥ 0.

• Metric fibration

  Given a complete metric space (X, d), two subsets M ₁ and M ₂ of X are called equidistant if for each x ∈ M ₁ there exists y ∈ M ₂ with d(x, y) being equal to the Hausdorff metric between the sets M ₁ and M ₂. A metric fibration of (X, d) is a partition $\mathcal{F}$ of X into isometric mutually equidistant closed sets.

  The quotient metric space $X/\mathcal{F}$ inherits a natural metric for which the distance map is a submetry.

• Homeomorphic metric spaces

  Two metric spaces (X, d _X ) and (Y, d _Y ) are called homeomorphic (or topologically isomorphic) if there exists a homeomorphism from X to Y, i.e., a bijective function f: X → Y such that f and f ⁻¹ are continuous (the preimage of every open set in Y is open in X).

  Two metric spaces (X, d _X ) and (Y, d _Y ) are called uniformly isomorphic if there exists a bijective function f: X → Y such that f and f ⁻¹ are uniformly continuous. A function g is uniformly continuous if, for any ε > 0, there exists δ > 0 such that, for any x, y ∈ X, the inequality d _X (x, y) < δ implies that d _Y (g(x), f(y)) < ε; a continuous function is uniformly continuous if X is compact.

• Möbius mapping

  Given distinct points x, y, z, w of a metric space (X, d), their cross-ratio is

  $$\displaystyle{cr((x,y,z,w),d) = \frac{d(x,y)d(z,w)} {d(x,z)d(y,w)} > 0.}$$

  Given metric spaces (X, d _X ) and (Y, d _Y ), a homeomorphism f: X → Y is called a Möbius mapping if, for every distinct points x, y, z, w ∈ X, it holds

  $$\displaystyle{cr((x,y,z,w),d_{X}) = cr((\,f(x),f(y),f(z),f(w)),d_{Y }).}$$

  A homeomorphism f: X → Y is called a quasi-Möbius mapping (Väisälä, 1984) if there exists a homeomorphism τ: [0, ∞) → [0, ∞) such that, for every quadruple x, y, z, w of distinct points of X, it holds

  $$\displaystyle{cr((\,f(x),f(y),f(z),f(w)),d_{Y }) \leq \tau (cr((x,y,z,w),d_{X})).}$$

  A metric space (X, d) is called metrically dense (or μ-dense for given μ > 1, Aseev–Trotsenko, 1987) if for any x, y ∈ X, there exists a sequence $\{z_{i},i \in \mathbb{Z}\}$ with z _i  → x as i → −∞, z _i  → y as i → ∞, and logcr((x, z _i , z _i+1, y), d) ≤ logμ for all $i \in \mathbb{Z}$. The space (X, d) is μ-dense if and only if (Tukia-Väisälä, 1980), for any x, y ∈ X, there exists z ∈ X with $\frac{d(x,y)} {6\mu } \leq d(x,z) \leq \frac{d(x,y)} {4}$.

• Quasi-symmetric mapping

  Given metric spaces (X, d _X ) and (Y, d _Y ), a homeomorphism f: X → Y is called a quasi-symmetric mapping (Tukia–Väisälä, 1980) if there is a homeomorphism τ: [0, ∞) → [0, ∞) such that, for every triple (x, y, z) of distinct points of X,

  $$\displaystyle{\frac{d_{Y }(\,f(x),f(y))} {d_{Y }(\,f(x),f(z))} \leq \tau \frac{d_{X}(x,y)} {d_{X}(x,z)}.}$$

  Quasi-symmetric mappings are quasi-Möbius, and quasi-Möbius mappings between bounded metric spaces are quasi-symmetric. In the case $f: \mathbb{R}^{n} \rightarrow \mathbb{R}^{n}$, quasi-symmetric mappings are exactly the same as quasi-conformal mappings.

• Conformal metric mapping

  Given metric spaces (X, d _X ) and (Y, d _Y ) which are domains in $\mathbb{R}^{n}$, a homeomorphism f: X → Y is called a conformal metric mapping if, for any nonisolated point x ∈ X, the limit $\lim _{y\rightarrow x}\frac{d_{Y }(\,f(x),f(y))} {d(x,y)}$ exists, is finite and positive.

  A homeomorphism f: X → Y is called a quasi-conformal mapping (or, specifically, C-quasi-conformal mapping) if there exists a constant C such that

  $$\displaystyle{\lim _{r\rightarrow 0}\sup \frac{\max \{d_{Y }(\,f(x),f(y)): d_{X}(x,y) \leq r\}} {\min \{d_{Y }(\,f(x),f(y)): d_{X}(x,y) \geq r\}} \leq C}$$

  for each x ∈ X. The smallest such constant C is called the conformal dilation.

  The conformal dimension of a metric space (X, d) (Pansu, 1989) is the infimum of the Hausdorff dimension over all quasi-conformal mappings of (X, d) into some metric space. For the middle-third Cantor set on [0, 1], it is 0 but, for any of its quasi-conformal images, it is positive.

• Hölder mapping

  Let c, α ≥ 0 be constants. Given metric spaces (X, d _X ) and (Y, d _Y ), a function f: X → Y is called the Hölder mapping (or α-Hölder mapping if the constant α should be mentioned) if for all x, y ∈ X

  $$\displaystyle{d_{Y }(\,f(x),f(y)) \leq c(d_{X}(x,y))^{\alpha }.}$$

  A 1-Hölder mapping is a Lipschitz mapping; 0-Hölder mapping means that the metric d _Y is bounded.

• Lipschitz mapping

  Let c be a positive constant. Given metric spaces (X, d _X ) and (Y, d _Y ), a function f: X → Y is called a Lipschitz (or Lipschitz continuous, c-Lipschitz if the constant c should be mentioned) mapping if for all x, y ∈ X it holds

  $$\displaystyle{d_{Y }(\,f(x),f(y)) \leq cd_{X}(x,y).}$$

  A c-Lipschitz mapping is called a metric mapping if c = 1, and is called a contraction if c < 1.

• Bi-Lipschitz mapping

  Given metric spaces (X, d _X ), (Y, d _Y ) and a constant c > 1, a function f: X → Y is called a bi-Lipschitz mapping (or c-bi-Lipschitz mapping, c-embedding ) if there exists a number r > 0 such that for any x, y ∈ X it holds

  $$\displaystyle{rd_{X}(x,y) \leq d_{Y }(\,f(x),f(y)) \leq crd_{X}(x,y).}$$

  Every bi-Lipschitz mapping is a quasi-symmetric mapping.

  The smallest c for which f is a c-bi-Lipschitz mapping is called the distortion of f. Bourgain, 1985, proved that every k-point metric space c-embeds into a Euclidean space with distortion O(lnk). Gromov’s distortion for curves is the maximum ratio of arc length to chord length.

  Two metrics d ₁ and d ₂ on X are called bi-Lipschitz equivalent metrics if there are positive constants c and C such that cd ₁(x, y) ≤ d ₂(x, y) ≤ Cd ₁(x, y) for all x, y ∈ X, i.e., the identity mapping is a bi-Lipschitz mapping from (X, d ₁) into (X, d ₂). Bi-Lipschitz equivalent metrics are equivalent, i.e., generate the same topology but, for example, equivalent L ₁-metric and L ₂-metric (cf. L _p -metric in Chap. 5) on $\mathbb{R}$ are not bi-Lipschitz equivalent.

  A bi-Lipschitz mapping f: X → Y is a c-isomorphism f: X → f(X).

• c-isomorphism of metric spaces

  Given two metric spaces (X, d _X ) and (Y, d _Y ), the Lipschitz norm | | . | |  _Lip on the set of all injective mappings f: X → Y is defined by

  $$\displaystyle{\vert \vert \,f\vert \vert _{Lip} =\sup _{x,y\in X,x\neq y}\frac{d_{Y }(\,f(x),f(y))} {d_{X}(x,y)}.}$$

  Two metric spaces X and Y are called c-isomorphic if there exists an injective mapping f: X → Y such that | | f | |  _Lip  | | f ⁻¹ | |  _Lip  ≤ c.

• Metric Ramsey number

  For a given class $\mathcal{M}$ of metric spaces (usually, l _p -spaces), an integer n ≥ 1, and a real number c ≥ 1, the metric Ramsey number (or c-metric Ramsey number) $R_{\mathcal{M}}(c,n)$ is the largest integer m such that every n-point metric space has a subspace of cardinality m that c-embeds into a member of $\mathcal{M}$ (see [BLMN05]).

  The Ramsey number R _n is the minimal number of vertices of a complete graph such that any edge-coloring with n colors produces a monochromatic triangle. The following metric analog of R _n was considered in [Masc04]: the least number of points a finite metric space must contain in order to contain an equilateral triangle, i.e., to have equilateral metric dimension greater than two.

• Uniform metric mapping

  Given metric spaces (X, d _X ) and (Y, d _Y ), a function f: X → Y is called a uniform metric mapping if there are two nondecreasing functions g ₁ and g ₂ from $\mathbb{R}_{\geq 0}$ to itself with lim _{r → ∞} g _i (r) = ∞ for i = 1, 2, such that the inequality

  $$\displaystyle{g_{1}(d_{X}(x,y)) \leq d_{Y }(\,f(x),f(y)) \leq g_{2}(d_{X}(x,y))}$$

  holds for all x, y ∈ X. A bi-Lipschitz mapping is a uniform metric mapping with linear functions g ₁, g ₂.

• Metric compression

  Given metric spaces (X, d _X ) (unbounded) and (Y, d _Y ), a function f: X → Y is a large scale Lipschitz mapping if, for some c > 0, D ≥ 0 and all x, y ∈ X,

  $$\displaystyle{d_{Y }(\,f(x),f(y)) \leq cd_{X}(x,y) + D.}$$

  The compression of such a mapping f is $\rho _{f}(r) =\inf _{d_{X}(x,y)\geq r}d_{Y }(\,f(x),f(y))$.

  The metric compression of (X, d _X ) in (Y, d _Y ) is defined by

  $$\displaystyle{R(X,Y ) =\sup _{f}\{\underline{\lim }_{r\rightarrow \infty }\frac{\log \max \{\rho _{f}(r),1\}} {\log r} \},}$$

  where the supremum is over all large scale Lipschitz mappings f.

  In the main interesting case—when (Y, d _Y ) is a Hilbert space and (X, d _X ) is a (finitely generated discrete) group with word metric —R(X, Y ) = 0 if there is no (Guentner–Kaminker, 2004) uniform metric mapping (X, d _X ) → (Y, d _Y ), and R(X, Y ) = 1 for free groups, even if there is no quasi-isometry. Arzhantzeva–Guba–Sapir, 2006, found groups with $\frac{1} {2} \leq R(X,Y ) \leq \frac{3} {4}$.

• Quasi-isometry

  Given metric spaces (X, d _X ) and (Y, d _Y ), a function f: X → Y is called a quasi-isometry (or (C,c)-quasi-isometry ) if it holds

  $$\displaystyle{C^{-1}d_{ X}(x,y) - c \leq d_{Y }(\,f(x),f(y)) \leq Cd_{X}(x,y) + c,}$$

  for some C ≥ 1, c ≥ 0, and $Y = \cup _{x\in X}B_{d_{Y }}(\,f(x),c)$, i.e., for every point y ∈ Y, there exists x ∈ X such that $d_{Y }(y,f(x)) < \frac{c} {2}$. Quasi-isometry is an equivalence relation on metric spaces; it is a bi-Lipschitz equivalence up to small distances. Quasi-isometry means that metric spaces contain bi-Lipschitz equivalent Delone sets.

  A quasi-isometry with C = 1 is called a coarse isometry (or rough isometry, almost isometry). Cf. quasi-Euclidean rank of a metric space.

• Coarse embedding

  Given metric spaces (X, d _X ) and (Y, d _Y ), a function f: X → Y is called a coarse embedding if there exist nondecreasing functions ρ ₁, ρ ₂: [0, ∞) → [0, ∞) with ρ ₁(d _X (x, x′)) ≤ d _Y ( f(x), f(x′)) ≤ ρ ₂(d _X (x, x′)) if x, x′ ∈ X and lim _{t → ∞} ρ ₁(t) = +∞.

  Metrics d ₁, d ₂ on X are called coarsely equivalent metrics if there exist nondecreasing functions f, g: [0, ∞) → [0, ∞) such that d ₁ ≤ f(d ₂), d ₂ ≤ g(d ₁).

• Metrically regular mapping

  Let (X, d _X ) and (Y, d _Y ) be metric spaces, and let F be a set-valued mapping from X to Y, having inverse F ⁻¹, i.e., with x ∈ F ⁻¹(y) if and only if y ∈ F(x).

  The mapping F is said to be metrically regular at $\overline{x}$ for $\overline{y}$ (Dontchev–Lewis–Rockafeller, 2002) if there exists c > 0 such that it holds

  $$\displaystyle{d_{X}(x,F^{-1}(y)) \leq cd_{ Y }(y,F(x))}$$

  for all (x, y) close to $(\overline{x},\overline{y})$. Here d(z, A) = inf _a ∈ A d(z, a) and d(z, ∅) = +∞.

• Contraction

  Given metric spaces (X, d _X ) and (Y, d _Y ), a function f: X → Y is called a contraction if the inequality

  $$\displaystyle{d_{Y }(\,f(x),f(y)) \leq cd_{X}(x,y)}$$

  holds for all x, y ∈ X and some real number c, 0 ≤ c < 1.

  Every contraction is a contractive mapping, and it is uniformly continuous. Banach fixed point theorem (or contraction principle): every contraction from a complete metric space into itself has a unique fixed point.

• Contractive mapping

  Given metric spaces (X, d _X ) and (Y, d _Y ), a function f: X → Y is called a contractive (or strictly short, distance-decreasing) mapping if

  $$\displaystyle{d_{Y }(\,f(x),f(y)) < d_{X}(x,y)}$$

  holds for all different x, y ∈ X. A function f: X → Y is called a noncontractive mapping (or dominating mapping) if for all x, y ∈ X it holds

  $$\displaystyle{d_{Y }(\,f(x),f(y)) \geq d_{X}(x,y).}$$

  Every noncontractive bijection from a totally bounded metric space onto itself is an isometry.

• Short mapping

  Given metric spaces (X, d _X ) and (Y, d _Y ), a function f: X → Y is called a short (or 1-Lipschitz, nonexpansive, distance-noninreasing, metric) mapping (or semicontraction) if for all x, y ∈ X it holds

  $$\displaystyle{d_{Y }(\,f(x),f(y)) \leq d_{X}(x,y).}$$

  A submetry is a short mapping such that the image of any metric ball is a metric ball of the same radius.

  The set of short mappings f: X → Y for bounded metric spaces (X, d _X ) and (Y, d _Y ) is a metric space under the uniform metric sup{d _Y ( f(x), g(x)): x ∈ X}.

  Two subsets A and B of a metric space (X, d) are called (Gowers, 2000) similar if there exist short mappings f: A → X, g: B → X and a small ε > 0 such that every point of A is within ε of some point of B, every point of B is within ε of some point of A, and | d(x, g( f(x))) − d(y, f(g(y))) | ≤ ε for any x ∈ A, y ∈ B.

• Category of metric spaces

  A category $\Psi$ consists of a class $Ob(\Psi )$ of objects and a class $Mor(\Psi )$ of morphisms (or arrows) satisfying the following conditions:

  • 1. To each ordered pair of objects A, B is associated a set $\Psi (A,B)$ of morphisms, and each morphism belongs to only one set $\Psi (A,B)$;

  • 2. The composition f ⋅ g of two morphisms f: A → B, g: C → D is defined if B = C in which case it belongs to $\Psi (A,D)$, and it is associative;

  • 3. Each set $\Psi (A,A)$ contains, as an identity, a morphism id _A such that f ⋅ id _A  = f and id _A ⋅ g = g for any morphisms f: X → A and g: A → Y.

  The category of metric spaces, denoted by Met (see [Isbe64]), is a category which has metric spaces as objects and short mappings as morphisms. A unique injective envelope exists in this category for every one of its objects; it can be identified with its tight span. In Met, the monomorphisms are injective short mappings, and isomorphisms are isometries. Met is a subcategory of the category which has metric spaces as objects and Lipschitz mappings as morphisms.

  Cf. metric 1-space on the objects of a category in Chap. 3

• Injective metric space

  A metric space (X, d) is called injective if, for every isometric embedding f: X → X′ of (X, d) into another metric space (X′, d′), there exists a short mapping f′ from X′ into X with f′ ⋅ f = id _X , i.e., X is a retract of X′.

  Equivalently, X is an absolute retract, i.e., a retract of every metric space into which it embeds isometrically. A metric space (X, d) is injective if and only if it is hyperconvex. Examples of such metric spaces are l ₁ ²-space, l _∞ ⁿ-space, any real tree and the tight span of a metric space.

• Injective envelope

  The injective envelope (introduced first in [Isbe64] as injective hull) is a generalization of Cauchy completion. Given a metric space (X, d), it can be embedded isometrically into an injective metric space $(\hat{X},\hat{d})$; given any such isometric embedding $f: X \rightarrow \hat{X}$, there exists a unique smallest injective subspace $(\overline{X},\overline{d})$ of $(\hat{X},\hat{d})$ containing f(X) which is called the injective envelope of X. It is isometrically identified with the tight span of (X, d).

  A metric space coincides with its injective envelope if and only if it is injective.

• Tight extension

  An extension (X′, d′) of a metric space (X, d) is called a tight extension if, for every semimetric d″ on X′ satisfying the conditions d″(x ₁, x ₂) = d(x ₁, x ₂) for all x ₁, x ₂ ∈ X, and d″(y ₁, y ₂) ≤ d′(y ₁, y ₂) for any y ₁, y ₂ ∈ X′, one has d″(y ₁, y ₂) = d′(y ₁, y ₂) for all y ₁, y ₂ ∈ X′.

  The tight span is the universal tight extension of X, i.e., it contains, up to isometries, every tight extension of X, and it has no proper tight extension itself.

• Tight span

  Given a metric space (X, d) of finite diameter, consider the set $\mathbb{R}^{X} =\{\, f: X \rightarrow \mathbb{R}\}$. The tight span T(X, d) of (X, d) is defined as the set $T(X,d) =\{\, f \in \mathbb{R}^{X}: f(x) =\sup _{y\in X}(d(x,y) - f(y))\mbox{ for all }x \in X\}$, endowed with the metric induced on T(X, d) by the sup norm | | f | |  = sup _x ∈ X  | f(x) | .

  The set X can be identified with the set {h _x  ∈ T(X, d): h _x (y) = d(y, x)} or, equivalently, with the set T ⁰(X, d) = { f ∈ T(X, d): 0 ∈ f(X)}. The injective envelope $(\overline{X},\overline{d})$ of X is isometrically identified with the tight span T(X, d) by

  $$\displaystyle{\overline{X} \rightarrow T(X,d),\,\,\overline{x} \rightarrow h_{\overline{x}} \in T(X,d): h_{\overline{x}}(y) = \overline{d}(\,f(y),\overline{x}).}$$

  The tight span T(X, d) of a finite metric space is the metric space (T(X), D( f, g) = max | f(x) − g(x) | ), where T(X) is the set of functions $f: X \rightarrow \mathbb{R}$ such that for any x, y ∈ X, f(x) + f(y) ≥ d(x, y) and, for each x ∈ X, there exists y ∈ X with f(x) + f(y) = d(x, y). The mapping of any x into the function f _x (y) = d(x, y) gives an isometric embedding of (X, d) into T(X, d). For example, if X = { x ₁, x ₂}, then T(X, d) is the interval of length d(x ₁, x ₂).

  The tight span of a metric space (X, d) of finite diameter can be considered as a polytopal complex of bounded faces of the polyhedron

  $$\displaystyle{\{y \in \mathbb{R}_{\geq 0}^{n}: y_{ i} + y_{j} \geq d(x_{i},x_{j})\,\,\mbox{ for}\,\,1 \leq i < j \leq n\}}$$

  if, for example, $X =\{ x_{1},\ldots,x_{n}\}$. The dimension of this complex is called (Dress, 1984) the combinatorial dimension of (X, d).

• Real tree

  A metric space (X, d) is called (Tits, 1977) a real tree (or ℝ-tree ) if, for all x, y ∈ X, there exists a unique arc from x to y, and this arc is a geodesic segment. So, an $\mathbb{R}$-tree is a (uniquely) arcwise connected metric space in which each arc is isometric to a subarc of $\mathbb{R}$. $\mathbb{R}$-tree is not related to a metric tree in Chap. 17

  A metric space (X, d) is a real tree if and only if it is path-connected and Gromov 0-hyperbolic (i.e., satisfies the four-point inequality ). The plane $\mathbb{R}^{2}$ with the Paris metric or lift metric (Chap. 19) are examples of an $\mathbb{R}$-tree.

  Real trees are exactly tree-like metric spaces which are geodesic; they are injective metric spaces among tree-like spaces. Tree-like metric spaces are by definition metric subspaces of real trees.

  If (X, d) is a finite metric space, then the tight span T(X, d) is a real tree and can be viewed as an edge-weighted graph-theoretical tree.

  A metric space is a complete real tree if and only if it is hyperconvex and any two points are joined by a metric segment.

  A geodesic metric space (X, d) is called (Druţu–Sapir, 2005) tree-graded with respect to a collection $\mathcal{P}$ of connected proper subsets with | P ∩ P′ | ≤ 1 for any distinct $P,P' \in \mathcal{P}$, if every its simple loop composed of three geodesics is contained in one $P \in \mathcal{P}$. $\mathbb{R}$-trees are tree-graded with respect to the empty set.

5 General Distances

• Discrete metric

  Given a set X, the discrete metric (or trivial metric , sorting distance , drastic distance , Dirac distance , overlap) is a metric on X, defined by d(x, y) = 1 for all distinct x, y ∈ X and d(x, x) = 0. Cf. the much more general notion of a (metrically or topologically) discrete metric space.

• Indiscrete semimetric

  Given a set X, the indiscrete semimetric d is a semimetric on X defined by d(x, y) = 0 for all x, y ∈ X.

• Equidistant metric

  Given a set X and a positive real number t, the equidistant metric d is a metric on X defined by d(x, y) = t for all distinct x, y ∈ X (and d(x, x) = 0).

• (1,2) − B-metric

  Given a set X, the (1,2) − B-metric d is a metric on X such that, for any x ∈ X, the number of points y ∈ X with d(x, y) = 1 is at most B, and all other distances are equal to 2. The (1,2) − B-metric is the truncated metric of a graph with maximal vertex degree B.

• Permutation metric

  Given a finite set X, a metric d on it is called a permutation metric (or linear arrangement metric) if there exists a bijection $\omega: X \rightarrow \{ 1,\ldots,\vert X\vert \}$ such that

  $$\displaystyle{d(x,y) = \vert \omega (x) -\omega (y)\vert }$$

  holds for all x, y ∈ X. Even–Naor–Rao–Schieber, 2000, defined a more general spreading metric , i.e., any metric d on $\{1,\ldots,n\}$ such that $\sum _{y\in M}d(x,y) \geq \frac{\vert M\vert (\vert M\vert +2)} {4}$ for any 1 ≤ x ≤ n and $M \subseteq \{1,\ldots,n\}\setminus \{x\}$ with | M | ≥ 2.

• Induced metric

  Given a metric space (X, d) and a subset X′ ⊂ X, an induced metric (or submetric ) is the restriction d′ of d to X′. A metric space (X′, d′) is called a metric subspace of (X, d), and (X, d) is called a metric extension of (X′, d′).

• Katĕtov mapping

  Given a metric space (X, d), the mapping $f: X \rightarrow \mathbb{R}$ is a Katĕtov mapping if

  $$\displaystyle{\vert \,f(x) - f(y)\vert \leq d(x,y) \leq f(x) + f(y)}$$

  for any x, y ∈ X, i.e., setting d(x, z) = f(x) defines a one-point metric extension (X ∪{ z}, d) of (X, d).

  The set E(X) of Katĕtov mappings on X is a complete metric space with metric D( f, g) = sup _x ∈ X  | f(x) − g(x) | ; (X, d) embeds isometrically in it via the Kuratowski mapping x → d(x, . ), with unique extension of each isometry of X to one of E(X).

• Dominating metric

  Given metrics d and d ₁ on a set X, d ₁ dominates d if d ₁(x, y) ≥ d(x, y) for all x, y ∈ X. Cf. noncontractive mapping (or dominating mapping).

• Barbilian semimetric

  Given sets X and P, the function $f: P \times X \rightarrow \mathbb{R}_{>0}$ is called an influence (of P over X) if for any x, y ∈ X the ratio $g_{xy}(\,p) = \frac{f(\,p,x)} {f(\,p,y)}$ has a maximum when p ∈ P.

  The Barbilian semimetric is defined on the set X by

  $$\displaystyle{\ln \frac{\max _{p\in P}g_{xy}(\,p)} {\min _{p\in P}g_{xy}(\,p)}}$$

  for any x, y ∈ X. Barbilian, 1959, proved that the above function is well defined (moreover, $\min _{p\in P}g_{xy}(\,p) = \frac{1} {\max _{p\in P}g_{yx}(\,p)}$) and is a semimetric. Also, it is a metric if the influence f is effective, i.e., there is no pair x, y ∈ X such that g _xy ( p) is constant for all p ∈ P. Cf. a special case Barbilian metric in Chap. 6.

• Metric transform

  A metric transform is a distance obtained as a function of a given metric (cf. Chap. 4).

• Complete metric

  Given a metric space (X, d), a sequence {x _n }, x _n  ∈ X, is said to have convergence to x ^∗ ∈ X if lim _{n → ∞} d(x _n , x ^∗) = 0, i.e., for any ε > 0, there exists $n_{0} \in \mathbb{N}$ such that d(x _n , x ^∗) < ε for any n > n ₀. Any sequence converges to at most one limit in X; it is not so, in general, if d is a semimetric.

  A sequence {x _n } _n , x _n  ∈ X, is called a Cauchy sequence if, for any ε > 0, there exists $n_{0} \in \mathbb{N}$ such that d(x _n , x _m ) < ε for any m, n > n ₀.

  A metric space (X, d) is called a complete metric space if every Cauchy sequence in it converges. In this case the metric d is called a complete metric. An example of an incomplete metric space is $(\mathbb{N},d(m,n) = \frac{\vert m-n\vert } {mn} )$.

• Cauchy completion

  Given a metric space (X, d), its Cauchy completion is a metric space (X ^∗, d ^∗) on the set X ^∗ of all equivalence classes of Cauchy sequences, where the sequence {x _n } _n is called equivalent to {y _n } _n if lim _{n → ∞} d(x _n , y _n ) = 0. The metric d ^∗ is defined by

  $$\displaystyle{d^{{\ast}}(x^{{\ast}},y^{{\ast}}) =\lim _{ n\rightarrow \infty }d(x_{n},y_{n}),}$$

  for any x ^∗, y ^∗ ∈ X ^∗, where {z _n } _n is any element in the equivalence class z ^∗.

  The Cauchy completion (X ^∗, d ^∗) is a unique, up to isometry, complete metric space, into which the metric space (X, d) embeds as a dense metric subspace.

  The Cauchy completion of the metric space $(\mathbb{Q},\vert x - y\vert )$ of rational numbers is the real line $(\mathbb{R},\vert x - y\vert )$. A Banach space is the Cauchy completion of a normed vector space (V, | | . | | ) with the norm metric | | x − y | | . A Hilbert space corresponds to the case an inner product norm $\vert \vert x\vert \vert = \sqrt{\langle x, x\rangle }$.

• Perfect metric space

  A complete metric space (X, d) is called perfect if every point x ∈ X is a limit point, i.e., | B(x, r) = { y ∈ X: d(x, y) < r} |  > 1 holds for any r > 0.

  A topological space is a Cantor space (i.e., homeomorphic to the Cantor set with the natural metric | x − y | ) if and only if it is nonempty, perfect, totally disconnected, compact and metrizable. The totally disconnected countable metric space $(\mathbb{Q},\vert x - y\vert )$ of rational numbers also consists only of limit points but it is not complete and not locally compact.

  Every proper metric ball of radius r in a metric space has diameter at most 2r. Given a number 0 < c ≤ 1, a metric space is called a c-uniformly perfect metric space if this diameter is at least 2cr. Cf. the radii of metric space.

• Metrically discrete metric space

  A metric space (X, d) is called metrically (or uniformly) discrete if there exists a number r > 0 such that B(x, r) = { y ∈ X: d(x, y) < r} = { x} for every x ∈ X.

  (X, d) is a topologically discrete metric space (or a discrete metric space) if the underlying topological space is discrete, i.e., each point x ∈ X is an isolated point: there exists a number r(x) > 0 such that B(x, r(x)) = { x}. For $X =\{ \frac{1} {n}: n = 1,2,3,\ldots \}$, the metric space (X, | x − y | ) is topologically but not metrically discrete. Cf. translation discrete metric in Chap. 10.

  Alternatively, a metric space (X, d) is called discrete if any of the following holds:

  • 1. (Burdyuk–Burdyuk 1991) it has a proper isolated subset, i.e., M ⊂ X with inf{d(x, y): x ∈ M, y ∉ M} > 0 (any such space admits a unique decomposition into continuous, i.e., nondiscrete, components);

  • 2. (Lebedeva–Sergienko–Soltan, 1984) for any distinct points x, y ∈ X, there exists a point z of the closed metric interval I(x, y) with I(x, z) = { x, z};

  • 3. a stronger property holds: for any two distinct points x, y ∈ X, every sequence of points $z_{1},z_{2},\ldots$ with z _k  ∈ I(x, y) but z _k+1 ∈ I(x, z _k )∖{z _k } for $k = 1,2,\ldots$ is a finite sequence.

• Locally finite metric space

  Let (X, d) be a metrically discrete metric space. Then it is called locally finite if for every x ∈ X and every r ≥ 0, the ball | B(x, r) | is finite.

  If, moreover, | B(x, r) | ≤ C(r) for some number C(r) depending only on r, then (X, d) is said to have bounded geometry.

• Bounded metric space

  A metric (moreover, a distance) d on a set X is called bounded if there exists a constant C > 0 such that d(x, y) ≤ C for any x, y ∈ X.

  For example, given a metric d on X, the metric D on X, defined by $D(x,y) = \frac{d(x,y)} {1+d(x,y)}$, is bounded with C = 1.

  A metric space (X, d) with a bounded metric d is called a bounded metric space.

• Totally bounded metric space

  A metric space (X, d) is called totally bounded if, for every ε > 0, there exists a finite ε -net, i.e., a finite subset M ⊂ X with the point-set distance d(x, M) < ε for any x ∈ X (cf. totally bounded space in Chap. 2).

  Every totally bounded metric space is bounded and separable. A metric space is totally bounded if and only if its Cauchy completion is compact.

• Separable metric space

  A metric space (X, d) is called separable if it contains a countable dense subset M, i.e., a subset with which all its elements can be approached: X is the closure c l(M) (M together with all its limit points).

  A metric space is separable if and only if it is second-countable (cf. Chap. 2).

• Compact metric space

  A compact metric space (or metric compactum ) is a metric space in which every sequence has a Cauchy subsequence, and those subsequences are convergent. A metric space is compact if and only if it is totally bounded and complete.

  Every bounded and closed subset of a Euclidean space is compact. Every finite metric space is compact. Every compact metric space is second-countable.

  A continuum is a nonempty connected metric compactum.

• Proper metric space

  A metric space is called proper (or finitely compact, having the Heine–Borel property) if every its closed metric ball is compact. Any such space is complete.

• UC metric space

  A metric space is called a UC metric space (or Atsuji space) if any continuous function from it into an arbitrary metric space is uniformly continuous.

  Every such space is complete. Every metric compactum is a UC metric space.

• Metric measure space

  A metric measure space (or mm-space, metric triple) is a triple (X, d, μ), where (X, d) is a Polish (i.e., complete separable; cf. Chap. 2) metric space and $(X,\Sigma,\mu )$ is a probability measure space (μ(X) = 1) with $\Sigma$ being a Borel σ-algebra of all open and closed sets of the metric topology (Chap. 2) induced by the metric d on X. Cf. metric outer measure.

• Norm metric

  Given a normed vector space (V, | | . | | ), the norm metric on V is defined by

  $$\displaystyle{\vert \vert x - y\vert \vert.}$$

  The metric space (V, | | x − y | | ) is called a Banach space if it is complete. Examples of norm metrics are l _p - and L _p -metrics, in particular, the Euclidean metric.

  Any metric space (X, d) admits an isometric embedding into a Banach space B such that its convex hull is dense in B (cf. Monge–Kantorovich metric in Chap. 14); (X, d) is a linearly rigid metric space if such an embedding is unique up to isometry. A metric space isometrically embeds into the unit sphere of a Banach space if and only if its diameter is at most 2.

• Path metric

  Given a connected graph G = (V, E), its path metric (or graphic metric) d _path is a metric on V defined as the length (i.e., the number of edges) of a shortest path connecting two given vertices x and y from V (cf. Chap. 15).

• Editing metric

  Given a finite set X and a finite set $\mathcal{O}$ of (unary) editing operations on X, the editing metric on X is the path metric of the graph with the vertex-set X and xy being an edge if y can be obtained from x by one of the operations from $\mathcal{O}$.

• Gallery metric

  A chamber system is a set X (its elements are called chambers) equipped with n equivalence relations ∼  _i , 1 ≤ i ≤ n. A gallery is a sequence of chambers x ₁, …, x _m such that x _i  ∼  _j x _i+1 for every i and some j depending on i.

  The gallery metric is an extended metric on X which is the length of the shortest gallery connecting x and y ∈ X (and is equal to ∞ if there is no connecting gallery). The gallery metric is the (extended) path metric of the graph with the vertex-set X and xy being an edge if x ∼  _i y for some 1 ≤ i ≤ n.

• Metric on incidence structure

  An incidence structure (P, L, I) consists of 3 sets: points P, lines L and flags I ⊂ P × L, where a point p ∈ P is said to be incident with a line l ∈ L if ( p, l) ∈ I.

  If, moreover, for any pair of distinct points, there is at most one line incident with both of them, then the collinearity graph is a graph whose vertices are the points with two vertices being adjacent if they determine a line.

  The metric on incidence structure is the path metric of this graph.

• Riemannian metric

  Given a connected n-dimensional smooth manifold M ⁿ (cf. Chaps. 2 and 7), its Riemannian metric is a collection of positive-definite symmetric bilinear forms ((g _ij )) on the tangent spaces of M ⁿ which varies smoothly from point to point.

  The length of a curve γ on M ⁿ is expressed as $\int _{\gamma }\sqrt{\sum _{i,j } g_{ij } dx_{i } dx_{j}}$, and the intrinsic metric on M ⁿ, also called the Riemannian distance, is the infimum of lengths of curves connecting any two given points x, y ∈ M ⁿ. Cf. Chap. 7

• Linearly additive metric

  A linearly additive (or additive on lines) metric is a continuous metric d on $\mathbb{R}^{n}$ which, for any points x, y, z lying in that order on a common line, satisfies

  $$\displaystyle{d(x,z) = d(x,y) + d(y,z).}$$

  Hilbert’s 4-th problem asked in 1900 to classify such metrics; it is solved only for dimension n = 2 ([Amba76]). Cf. projective metric in Chap. 6.

  Every norm metric on $\mathbb{R}^{n}$ is linearly additive. Every linearly additive metric on $\mathbb{R}^{2}$ is a hypermetric.

• Hamming metric

  The Hamming metric d _H (called sometimes Dalal distance in Semantics) is a metric on $\mathbb{R}^{n}$ defined (Hamming, 1950) by

  $$\displaystyle{\vert \{i: 1 \leq i \leq n,x_{i}\neq y_{i}\}\vert.}$$

  On binary vectors x, y ∈ { 0, 1}ⁿ the Hamming metric and the l ₁ -metric (cf. L _p -metric in Chap. 5) coincide; they are equal to $\vert I(x)\Delta I(y)\vert = \vert I(x)\setminus I(y)\vert + \vert I(y)\setminus I(x)\vert$, where I(z) = { 1 ≤ t ≤ n: z _i  = 1}.

  In fact, max{ | I(x)∖ I(y) | , | I(y)∖ I(x) | } is also a metric.

• Lee metric

  Given $m,n \in \mathbb{N}$, m ≥ 2, the Lee metric d _Lee is a metric on $\mathbb{Z}_{m}^{n} =\{ 0,1,\ldots,m - 1\}^{n}$ defined (Lee, 1958) by

  $$\displaystyle{\sum _{1\leq i\leq n}\min \{\vert x_{i} - y_{i}\vert,m -\vert x_{i} - y_{i}\vert \}.}$$

  The metric space (Z _m ⁿ, d _Lee ) is a discrete analog of the elliptic space.

  The Lee metric coincides with the Hamming metric d _H if m = 2 or m = 3. The metric spaces (Z ₄ ⁿ, d _Lee ) and Z ₂ ²ⁿ, d _H ) are isometric. Lee and Hamming metrics are applied for phase and orthogonal modulation, respectively.

  Cf. absolute summation distance and generalized Lee metric in Chap. 16.

• Enomoto–Katona metric

  Given a finite set X and an integer k, 2k ≤ | X | , the Enomoto–Katona metric (2001) is the distance between unordered pairs (X ₁, X ₂) and (Y ₁, Y ₂) of disjoint k-subsets of X defined by

  $$\displaystyle{\min \{\vert X_{1}\setminus Y _{1}\vert + \vert X_{2}\setminus Y _{2}\vert,\vert X_{1}\setminus Y _{2}\vert + \vert X_{2}\setminus Y _{1}\vert \}.}$$

  Cf. Earth Mover’s distance, transportation distance in Chaps. 21 and 14.

• Symmetric difference metric

  Given a measure space $(\Omega,\mathcal{A},\mu )$, the symmetric difference (or measure) semimetric on the set $\mathcal{A}_{\mu } =\{ A \in \mathcal{A}:\mu (A) < \infty \}$ is defined by

  $$\displaystyle{od_{\bigtriangleup }(A,B) =\mu (A\bigtriangleup B),}$$

  where A△B = (A ∪ B)∖(A ∩ B) is the symmetric difference of A and $B \in \mathcal{A}_{\mu }$.

  The value d _△(A, B) = 0 if and only if μ(A△B) = 0, i.e., A and B are equal almost everywhere. Identifying two sets $A,B \in \mathcal{A}_{\mu }$ if μ(A△B) = 0, we obtain the symmetric difference metric (or Fréchet–Nikodym–Aronszyan distance, measure metric ).

  If μ is the cardinality measure, i.e., μ(A) =  | A | , then d _△(A, B) =  | A△B |  =  | A ∖ B | + | B ∖ A | . In this case | A△B |  = 0 if and only if A = B.

  The metrics d _max(A, B) = max( | A ∖ B | , | B ∖ A | ) and $1 - \frac{\vert A\cap B\vert } {\max (\vert A\vert,\vert B\vert )}$ (its normalised version) are special cases of Zelinka distance and Bunke–Shearer metric in Chap. 15 For each p ≥ 1, the p-difference metric (Noradam–Nyblom, 2014) is $d_{p}(A,B) = (\vert A\setminus B\vert ^{p} + \vert B\setminus A\vert ^{p})^{\frac{1} {p} }$; so, d ₁ = d _△ and lim _{p → ∞} d _p  = d _max.

  The Johnson distance between k-sets A and B is $\frac{\vert A\bigtriangleup B\vert } {2} = k -\vert A \cap B\vert$.

  The symmetric difference metric between ordered q-partitions A = (A ₁, …, A _q ) and B = (B ₁, …, B _q ) is $\sum _{i=1}^{q}\vert A_{i}\Delta B_{i}\vert$. Cf. metrics between partitions in Chap. 10.

• Steinhaus distance

  Given a measure space $(\Omega,\mathcal{A},\mu )$, the Steinhaus distance d _St is a semimetric on the set $\mathcal{A}_{\mu } =\{ A \in \mathcal{A}:\mu (A) < \infty \}$ defined as 0 if μ(A) = μ(B) = 0, and by

  $$\displaystyle{ \frac{\mu (A\bigtriangleup B)} {\mu (A \cup B)} = 1 -\frac{\mu (A \cap B)} {\mu (A \cup B)}}$$

  if μ(A ∪ B) > 0. It becomes a metric on the set of equivalence classes of elements from $\mathcal{A}_{\mu }$; here $A,B \in \mathcal{A}_{\mu }$ are called equivalent if μ(A△B) = 0.

  The biotope (or Tanimoto ) distance $\frac{\vert A\bigtriangleup B\vert } {\vert A\cup B\vert }$ is the special case of Steinhaus distance obtained for the cardinality measure μ(A) =  | A | for finite sets.

  Cf. also the generalized biotope transform metric in Chap. 4.

• Fréchet metric

  Let (X, d) be a metric space. Consider a set $\mathcal{F}$ of all continuous mappings f: A → X, g: B → X, $\ldots$, where $A,B,\ldots$ are subsets of $\mathbb{R}^{n}$, homeomorphic to [0, 1]ⁿ for a fixed dimension $n \in \mathbb{N}$.

  The Fréchet semimetric d _F is a semimetric on $\mathcal{F}$ defined by

  $$\displaystyle{\inf _{\sigma }\sup _{x\in A}d(\,f(x),g(\sigma (x))),}$$

  where the infimum is taken over all orientation preserving homeomorphisms σ: A → B. It becomes the Fréchet metric on the set of equivalence classes f ^∗ = { g: d _F (g, f) = 0}. Cf. the Fréchet surface metric in Chap. 8

• Hausdorff metric

  Given a metric space (X, d), the Hausdorff metric (or two-sided Hausdorff distance) is a metric on the family $\mathcal{F}$ of nonempty compact subsets of X defined by

  $$\displaystyle{d_{Haus} =\max \{ d_{dHaus}(A,B),d_{dHaus}(B,A)\},}$$

  where d _dHaus (A, B) = max _x ∈ A min _y ∈ B d(x, y) is the directed Hausdorff distance (or one-sided Hausdorff distance) from A to B. The metric space $(\mathcal{F},d_{Haus})$ is called hyperspace of metric space (X, d); cf. hyperspace in Chap. 2

  In other words, d _Haus (A, B) is the minimal number ε (called also the Blaschke distance ) such that a closed ε -neighborhood of A contains B and a closed ε-neighborhood of B contains A. Then d _Haus (A, B) is equal to

  $$\displaystyle{\sup _{x\in X}\vert d(x,A) - d(x,B)\vert,}$$

  where d(x, A) = min _y ∈ A d(x, y) is the point-set distance.

  If the above definition is extended for noncompact closed subsets A and B of X, then d _Haus (A, B) can be infinite, i.e., it becomes an extended metric.

  For not necessarily closed subsets A and B of X, the Hausdorff semimetric between them is defined as the Hausdorff metric between their closures. If X is finite, d _Haus is a metric on the class of all subsets of X.

• L _p -Hausdorff distance

  Given a finite metric space (X, d), the L _p -Hausdorff distance ([Badd92]) between two subsets A and B of X is defined by

  $$\displaystyle{(\sum _{x\in X}\vert d(x,A) - d(x,B)\vert ^{p})^{\frac{1} {p} },}$$

  where d(x, A) is the point-set distance. The usual Hausdorff metric corresponds to the case p = ∞.

• Generalized G-Hausdorff metric

  Given a group (G, ⋅ , e) acting on a metric space (X, d), the generalized G-Hausdorff metric between two closed bounded subsets A and B of X is

  $$\displaystyle{\min _{g_{1},g_{2}\in G}d_{Haus}(g_{1}(A),g_{2}(B)),}$$

  where d _Haus is the Hausdorff metric. If d(g(x), g(y)) = d(x, y) for any g ∈ G (i.e., if the metric d is left-invariant with respect of G), then above metric is equal to min _g ∈ G d _Haus (A, g(B)).

• Gromov–Hausdorff metric

  The Gromov–Hausdorff metric is a metric on the set of all isometry classes of compact metric spaces defined by

  $$\displaystyle{\inf d_{Haus}(\,f(X),g(Y ))}$$

  for any two classes X ^∗ and Y ^∗ with the representatives X and Y, respectively, where d _Haus is the Hausdorff metric, and the minimum is taken over all metric spaces M and all isometric embeddings f: X → M, g: Y → M. The corresponding metric space is called the Gromov–Hausdorff space.

  The Hausdorff–Lipschitz distance between isometry classes of compact metric spaces X and Y is defined by

  $$\displaystyle{\inf \{d_{GH}(X,X_{1}) + d_{L}(X_{1},Y _{1}) + d_{GH}(Y,Y _{1})\},}$$

  where d _GH is the Gromov–Hausdorff metric, d _L is the Lipschitz metric, and the minimum is taken over all (isometry classes of compact) metric spaces X ₁, Y ₁.

• Kadets distance

  The gap (or opening) between two closed subspaces X and Y of a Banach space (V, | | . | | ) is defined by

  $$\displaystyle{gap(X,Y ) =\max \{\delta (X,Y ),\delta (Y,X)\},}$$

  where δ(X, Y ) = sup{inf _y ∈ Y  | | x − y | | : x ∈ X, | | x | |  = 1} (cf. gap distance in Chap. 12 and gap metric in Chap. 18).

  The Kadets distance between two Banach spaces V and W is a semimetric defined (Kadets, 1975) by

  $$\displaystyle{\inf _{Z,f,g}gap(\overline{B}_{f(V )},\overline{B}_{g(W)}),}$$

  where the infimum is taken over all Banach spaces Z and all linear isometric embeddings f: V → Z and g: W → Z; here $\overline{B}_{f(V )}$ and $\overline{B}_{g(W)}$ are the closed unit balls of Banach spaces f(V ) and g(W), respectively.

  The nonlinear analog of the Kadets distance is the following Gromov–Hausdorff distance between Banach spaces U and W:

  $$\displaystyle{\inf _{Z,f,g}d_{Haus}(\,f(\overline{B}_{V }),g(\overline{B}_{W})),}$$

  where the infimum is taken over all metric spaces Z and all isometric embeddings f: V → Z and g: W → Z; here d _Haus is the Hausdorff metric.

  The Kadets path distance between Banach spaces V and W is defined (Ostrovskii, 2000) as the infimum of the lengths (with respect to the Kadets distance) of all curves joining V and W (and is equal to ∞ if there is no such curve).

• Banach–Mazur distance

  The Banach–Mazur distance d _BM between two Banach spaces V and W is

  $$\displaystyle{\ln \inf _{T}\vert \vert T\vert \vert \cdot \vert \vert T^{-1}\vert \vert,}$$

  where the infimum is taken over all isomorphisms T: V → W.

  It can also be written as lnd(V, W), where the number d(V, W) is the smallest positive d ≥ 1 such that $\overline{B}_{W}^{n} \subset T(\overline{B}_{V }^{n}) \subset d\,\overline{B}_{W}^{n}$ for some linear invertible transformation T: V → W. Here $\overline{B}_{V }^{n} =\{ x \in V: \vert \vert x\vert \vert _{V } \leq 1\}$ and $\overline{B}_{W}^{n} =\{ x \in W;\vert \vert x\vert \vert _{W} \leq 1\}$ are the unit balls of the normed spaces (V, | | . | |  _V ) and (W, | | . | |  _W ), respectively.

  One has d _BM (V, W) = 0 if and only if V and W are isometric, and d _BM becomes a metric on the set X ⁿ of all equivalence classes of n-dimensional normed spaces, where V ∼ W if they are isometric. The pair (X ⁿ, d _BM ) is a compact metric space which is called the Banach–Mazur compactum.

  The modified Banach–Mazur distance (Glushkin, 1963, and Khrabrov, 2001) is

  $$\displaystyle{\inf \{\vert \vert T\vert \vert _{X\rightarrow Y }: \vert detT\vert = 1\} \cdot \inf \{\vert \vert T\vert \vert _{Y \rightarrow X}: \vert detT\vert = 1\}.}$$

  The weak Banach–Mazur distance (Tomczak–Jaegermann, 1984) is

  $$\displaystyle{\max \{\overline{\gamma }_{Y }(id_{X}),\overline{\gamma }_{X}(id_{Y })\},}$$

  where id is the identity map and, for an operator U: X → Y, $\overline{\gamma }_{Z}(U)$ denotes inf∑ | | W _k  | | ⋅ | | V _k  | | . Here the infimum is taken over all representations U = ∑ W _k V _k for W _k : X → Z and V _k : Z → Y. This distance never exceeds the corresponding Banach–Mazur distance.

• Lipschitz distance

  Given α ≥ 0 and two metric spaces (X, d _X ), (Y, d _Y ), the α-Hölder norm | | . | |  _Hol on the set of all injective functions f: X → Y is defined by

  $$\displaystyle{\vert \vert \,f\vert \vert _{Hol} =\sup _{x,y\in X,x\neq y}\frac{d_{Y }(\,f(x),f(y))} {d_{X}(x,y)^{\alpha }}.}$$

  The Lipschitz norm | | . | |  _Lip is the case α = 1 of | | . | |  _Hol .

  The Lipschitz distance between metric spaces (X, d _X ) and (Y, d _Y ) is defined by

  $$\displaystyle{\ln \inf _{f}\vert \vert \,f\vert \vert _{Lip} \cdot \vert \vert \,f^{-1}\vert \vert _{ Lip},}$$

  where the infimum is taken over all bijective functions f: X → Y. Equivalently, it is the infimum of numbers lna such that there exists a bijective bi-Lipschitz mapping between (X, d _X ) and (Y, d _Y ) with constants exp(−a), exp(a).

  It becomes a metric (Lipschitz metric ) on the set of all isometry classes of compact metric spaces. Cf. Hausdorff–Lipschitz distance.

  This distance is an analog to the Banach–Mazur distance and, in the case of finite-dimensional real Banach spaces, coincides with it.

  It also coincides with the Hilbert projective metric on nonnegative projective spaces, obtained by starting with $\mathbb{R}_{>0}^{n}$ and identifying any point x with cx, c > 0.

• Lipschitz distance between measures

  Given a compact metric space (X, d), the Lipschitz seminorm | | . | |  _Lip on the set of all functions $f: X \rightarrow \mathbb{R}$ is defined by $\vert \vert \,f\vert \vert _{Lip} =\sup _{x,y\in X,x\neq y}\frac{\vert \,f(x)-f(y)\vert } {d(x,y)}$.

  The Lipschitz distance between measures μ and ν on X is defined by

  $$\displaystyle{\sup _{\vert \vert \,f\vert \vert _{Lip}\leq 1}\int fd(\mu -\nu ).}$$

  It is the transportation distance (Chap. 14) if μ, ν are probability measures. Let a such measure m _x (. ) be attached to any x ∈ X; for distinct x, y the coarse Ricci curvature along (xy) is defined (Ollivier, 2009) as $\kappa (x,y) = 1 -\frac{W_{1}(m_{x},m_{y})} {d(x,y)}$. Ollivier’s curvature generalizes the Ricci curvature in Riemannian space (cf. Chap. 7).

• Barycentric metric space

  Given a metric space (X, d), let (B(X), | | μ −ν | |  _TV ) be the metric space, where B(X) is the set of all regular Borel probability measures on X with bounded support, and | | μ −ν | |  _TV is the variational distance ∫ _X  | p(μ) − p(ν) | dλ (cf. Chap. 14). Here p(μ) and p(ν) are the density functions of measures μ and ν, respectively, with respect to the σ-finite measure $\frac{\mu +\nu } {2}$.

  A metric space (X, d) is barycentric if there exists a constant β > 0 and a surjection f: B(X) → X such that for any measures μ, ν ∈ B(X) it holds the inequality

  $$\displaystyle{d(\,f(\mu ),f(\nu )) \leq \beta diam(supp(\mu +\nu ))\vert \vert \mu -\nu \vert \vert _{TV }.}$$

  Any Banach space (X, d =  | | x − y | | ) is a barycentric metric space with the smallest β being 1 and the map f(μ) being the usual center of mass ∫ _X xdμ(x).

  Any Hadamard (i.e., a complete CAT(0 ) space, cf. Chap. 6, is barycentric with the smallest β being 1 and the map f(μ) being the unique minimizer of the function g(y) = ∫ _X d ²(x, y)dμ(x) on X.

• Point-set distance

  Given a metric space (X, d), the point-set distance d(x, A) between a point x ∈ X and a subset A of X is defined as

  $$\displaystyle{\inf _{y\in A}d(x,y).}$$

  For any x, y ∈ X and for any nonempty subset A of X, we have the following version of the triangle inequality: d(x, A) ≤ d(x, y) + d(y, A) (cf. distance map ).

  For a given point-measure μ(x) on X and a penalty function p, an optimal quantizer is a set B ⊂ X such that ∫ p(d(x, B))dμ(x) is as small as possible.

• Set-set distance

  Given a metric space (X, d), the set-set distance between two subsets A and B of X is defined by

  $$\displaystyle{d_{ss}(A,B) =\inf _{x\in A,y\in B}d(x,y).}$$

  This distance can be 0 even for disjoint sets, for example, for the intervals (1, 2), (2, 3) on $\mathbb{R}$. The sets A and B are positively separated if d _ss (A, B) > 0. A constructive appartness space is a generalization of this relation on subsets of X.

  The spanning distance between A and B is sup _{x ∈ A, y ∈ B} d(x, y).

  In Data Analysis, (cf. Chap. 17) the set-set and spanning distances between clusters are called the single and complete linkage, respectively.

• Matching distance

  Given a metric space (X, d), the matching distance (or multiset-multiset distance) between two multisets A and B in X is defined by

  $$\displaystyle{\inf _{\phi }\max _{x\in A}d(x,\phi (x)),}$$

  where ϕ runs over all bijections between A and B, as multisets.

  The matching distance is not related to the perfect matching distance in Chap. 15 and to the nonlinear elastic matching distance in Chap. 21. But the bottleneck distance in Chap. 21 is a special case of it.

• Metrics between multisets

  A multiset (or bag) drawn from a set S is a mapping $m: S \rightarrow \mathbb{Z}_{\geq 0}$, where m(x) represents the “multiplicity” of x ∈ S. The dimensionality, cardinality and height of multiset m is | S | , | m |  = ∑ _x ∈ S m(x) and max _x ∈ S m(x), respectively.

  Multisets are good models for multi-attribute objects such as, say, all symbols in a string, all words in a document, etc.

  A multiset m is finite if S and all m(x) are finite; the complement of a finite multiset m is the multiset $\overline{m}: S \rightarrow \mathbb{Z}_{\geq 0}$, where $\overline{m}(x) =\max _{y\in S}m(y) - m(x)$. Given two multisets m ₁ and m ₂, denote by m ₁ ∪ m ₂, m ₁ ∩ m ₂, m ₁∖m ₂ and $m_{1}\Delta m_{2}$ the multisets on S defined, for any x ∈ S, by m ₁ ∪ m ₂(x) = max{m ₁(x), m ₂(x)}, m ₁ ∩ m ₂(x) = min{m ₁(x), m ₂(x)}, m ₁∖m ₂(x) = max{0, m ₁(x) − m ₂(x)} and $m_{1}\Delta m_{2}(x) = \vert m_{1}(x) - m_{2}(x)\vert$, respectively. Also, m ₁ ⊆ m ₂ denotes that m ₁(x) ≤ m ₂(x) for all x ∈ S.

  The measure μ(m) of a multiset m is a linear combination μ(m) = ∑ _x ∈ S λ(x)m(x) with λ(x) ≥ 0. In particular, | m | is the counting measure.

  For any measure $\mu (m) \in \mathbb{R}_{\geq 0}$, Miyamoto, 1990, and Petrovsky, 2003, proposed several semimetrics between multisets m ₁ and m ₂ including $d_{1}(m_{1},m_{2}) =\mu (m_{1}\Delta m_{2})$ and $d_{2}(m_{1},m_{2}) = \frac{\mu (m_{1}\Delta m_{2})} {\mu (m_{1}\cup m_{2})}$ (with d ₂(∅, ∅) = 0 by definition). Cf. symmetric difference metric and Steinhaus distance.

  Among examples of other metrics between multisets are matching distance, metric space of roots in Chap. 12, μ -metric in Chap. 15 and, in Chap. 11, bag distance max{ | m ₁∖m ₂ | , | m ₂∖m ₁ | } and q-gram similarity.

  See also Vitanyi multiset metric in Chap. 3

• Metrics between fuzzy sets

  A fuzzy subset of a set S is a mapping μ: S → [0, 1], where μ(x) represents the “degree of membership” of x ∈ S. It is an ordinary (crisp) if all μ(x) are 0 or 1. Fuzzy sets are good models for gray scale images (cf. gray scale images distances in Chap. 21), random objects and objects with nonsharp boundaries.

  Bhutani–Rosenfeld, 2003, introduced the following two metrics between two fuzzy subsets μ and ν of a finite set S. The diff-dissimilarity is a metric (a fuzzy generalization of Hamming metric ), defined by

  $$\displaystyle{d(\mu,\nu ) =\sum _{x\in S}\vert \mu (x) -\nu (x)\vert.}$$

  The perm-dissimilarity is a semimetric defined by

  $$\displaystyle{\min \{d(\mu,p(\nu ))\},}$$

  where the minimum is taken over all permutations p of S.

  The Chaudhuri–Rosenfeld metric (1996) between two fuzzy sets μ and ν with crisp points (i.e., the sets {x ∈ S: μ(x) = 1} and {x ∈ S: ν(x) = 1} are nonempty) is an extended metric, defined the Hausdorff metric d _Haus by

  $$\displaystyle{\int _{0}^{1}2td_{ Haus}(\{x \in S:\mu (x) \geq t\},\{x \in S:\nu (x) \geq t\})dt.}$$

  A fuzzy number is a fuzzy subset μ of the real line $\mathbb{R}$, such that the level set (or t-cut) $A_{\mu }(t) =\{ x \in \mathbb{R}:\mu (x) \geq t\}$ is convex for every t ∈ [0, 1]. The sendograph of a fuzzy set μ is the set send(μ) = { (x, t) ∈ S × [0, 1]: μ(x) > 0,  μ(x) ≥ t}. The sendograph metric (Kloeden, 1980) between two fuzzy numbers μ, ν with crisp points and compact sendographs is the Hausdorff metric

  $$\displaystyle{\max \{\sup _{a=(x,t)\in send(\mu )}d(a,send(\nu )),\sup _{b=(x',t')\in send(\nu )}d(b,send(\mu ))\},}$$

  where d(a, b) = d((x, t), (x′, t′)) is a box metric (Chap. 4) max{ | x − x′ | , | t − t′ | }.

  The Klement–Puri–Ralesku metric (1988) between fuzzy numbers μ, ν is

  $$\displaystyle{\int _{0}^{1}d_{ Haus}(A_{\mu }(t),A_{\nu }(t))dt,}$$

  where d _Haus (A _μ (t), A _ν (t)) is the Hausdorff metric

  $$\displaystyle{\max \{\sup _{x\in A_{\mu }(t)}\inf _{y\in A_{\nu }(t)}\vert x - y\vert,\sup _{x\in A_{\nu }(t)}\inf _{x\in A_{\mu }(t)}\vert x - y\vert \}.}$$

  Several other Hausdorff-like metrics on some families of fuzzy sets were proposed by Boxer in 1997, Fan in 1998 and Brass in 2002; Brass also argued the nonexistence of a “good” such metric.

  If q is a quasi-metric on [0, 1] and S is a finite set, then Q(μ, ν) = sup _x ∈ S q(μ(x), ν(x)) is a quasi-metric on fuzzy subsets of S.

  Cf. fuzzy Hamming distance in Chap. 11 and, in Chap. 23, fuzzy set distance and fuzzy polynucleotide metric. Cf. fuzzy metric spaces in Chap. 3 for fuzzy-valued generalizations of metrics and, for example, [Bloc99] for a survey.

• Metrics between intuitionistic fuzzy sets

  An intuitionistic fuzzy subset of a set S is (Atanassov, 1999) an ordered pair of mappings μ, ν: → [0, 1] with 0 ≤ μ(x) +ν(x) ≤ 1 for all x ∈ S, representing the “degree of membership” and the “degree of nonmembership” of x ∈ S, respectively. It is an ordinary fuzzy subset if μ(x) +ν(x) = 1 for all x ∈ S.

  Given two intuitionistic fuzzy subsets (μ(x), ν(x)) and (μ′(x), ν′(x)) of a finite set $S =\{ x_{1},\ldots,x_{n}\}$, their Atanassov distances (1999) are:

  $$\displaystyle{\frac{1} {2}\sum _{i=1}^{n}(\vert \mu (x_{ i}) -\mu '(x_{i})\vert + \vert \nu (x_{i}) -\nu '(x_{i})\vert )\mbox{ (Hamming distance) }}$$

  and, in general, for any given numbers p ≥ 1 and 0 ≤ q ≤ 1, the distance

  $$\displaystyle{(\sum _{i=1}^{n}(1 - q)(\mu (x_{ i}) -\mu '(x_{i}))^{p} + q(\nu (x_{ i}) -\nu '(x_{i})^{p})^{\frac{1} {p} }.}$$

  Their Grzegorzewski distances (2004) are:

  $$\displaystyle\begin{array}{rcl} & \sum _{i=1}^{n}\max \{\vert \mu (x_{i}) -\mu '(x_{i})\vert,\vert \nu (x_{i}) -\nu '(x_{i})\vert \}\mbox{ (Hamming distance); } & {}\\ & \sqrt{\sum _{i=1 }^{n }\max \{(\mu (x_{i } ) -\mu '(x_{i } ))^{2 }, (\nu (x_{i } ) -\nu '(x_{i } ))^{2}\}}\mbox{ (Euclidean distance)}.& {}\\ \end{array}$$

  The normalized versions (dividing the above sums by n) were also proposed.

  Szmidt–Kacprzyk, 1997, proposed a modification of the above, adding π(x) −π′(x), where π(x) is the third mapping 1 −μ(x) −ν(x).

  An interval-valued fuzzy subset of a set S is a mapping μ: → [I], where [I] is the set of closed intervals [a ⁻, a ⁺] ⊆ [0, 1]. Let μ(x) = [μ ⁻(x), μ ⁺(x)], where 0 ≤ μ ⁻(x) ≤ μ ⁺(x) ≤ 1 and an interval-valued fuzzy subset is an ordered pair of mappings μ ⁻ and μ ⁺. This notion is close to the above intuitionistic one; so, above distance can easily be adapted. For example, ∑ _i = 1 ⁿmax{ | μ ⁻(x _i ) −μ′⁻(x _i ) | , | μ ⁺(x _i ) −μ′⁺(x _i ) | } is a Hamming distance between interval-valued fuzzy subsets (μ ⁻, μ ⁺) and (μ′⁻, μ′⁺).

• Polynomial metric space

  Let (X, d) be a metric space with a finite diameter D and a finite normalized measure μ _X . Let the Hilbert space L ₂(X, d) of complex-valued functions decompose into a countable (when X is infinite) or a finite (with D + 1 members when X is finite) direct sum of mutually orthogonal subspaces $L_{2}(X,d) = V _{0} \oplus V _{1}\oplus \ldots$.

  Then (X, d) is a polynomial metric space if there exists an ordering of the spaces $V _{0},V _{1},\ldots$ such that, for $i = 0,1,\ldots$, there exist zonal spherical functions, i.e., real polynomials Q _i (t) of degree i such that

  $$\displaystyle{Q_{i}(t(d(x,y))) = \frac{1} {r_{i}}\sum _{j=1}^{r_{i} }v_{ij}(x)\overline{v_{ij}(y)}}$$

  for all x, y ∈ X, where r _i is the dimension of V _i , {v _ii (x): 1 ≤ j ≤ r _i } is an orthonormal basis of V _i , and t(d) is a continuous decreasing real function such that t(0) = 1 and t(D) = −1. The zonal spherical functions constitute an orthogonal system of polynomials with respect to some weight w(t).

  The finite polynomial metric spaces are also called (P and Q)-polynomial association schemes; cf. distance-regular graph in Chap. 15 The infinite polynomial metric spaces are the compact connected two-point homogeneous spaces. Wang, 1952, classified them as the Euclidean unit spheres, the real, complex, quaternionic projective spaces or the Cayley projective line and plane.

• Universal metric space

  A metric space (U, d) is called universal for a collection $\mathcal{M}$ of metric spaces if any metric space (M, d _M ) from $\mathcal{M}$ is isometrically embeddable in (U, d), i.e., there exists a mapping f: M → U which satisfies d _M (x, y) = d( f(x), f(y)) for any x, y ∈ M. Some examples follow.

  Every separable metric space (X, d) isometrically embeds (Fréchet, 1909) in (a nonseparable) Banach space l _∞ ^∞. In fact, d(x, y) = sup _i  | d(x, a _i ) − d(y, a _i ) | , where $(a_{1},\ldots,a_{i},\ldots )$ is a dense countable subset of X.

  Every metric space isometrically embeds (Kuratowski, 1935) in the Banach space L ^∞(X) of bounded functions $f: X \rightarrow \mathbb{R}$ with the norm sup _x ∈ X  | f(x) | .

  The Urysohn space is a homogeneous complete separable space which is the universal metric space for all separable metric spaces. The Hilbert cube (Chap. 4) is the universal space for the class of metric spaces with a countable base.

  The graphic metric space of the random graph (Rado, 1964; the vertex-set consists of all prime numbers $p \equiv 1\,(\mod \,4)$ with pq being an edge if p is a quadratic residue modulo q) is the universal metric space for any finite or countable metric space with distances 0, 1 and 2 only. It is a discrete analog of the Urysohn space.

  There exists a metric d on $\mathbb{R}$, inducing the usual (interval) topology, such that $(\mathbb{R},d)$ is a universal metric space for all finite metric spaces (Holsztynski, 1978). The Banach space l _∞ ⁿ is a universal metric space for all metric spaces (X, d) with | X | ≤ n + 2 (Wolfe, 1967). The Euclidean space $\mathbb{E}^{n}$ is a universal metric space for all ultrametric spaces (X, d) with | X | ≤ n + 1; the space of all finite functions $f(t): \mathbb{R}_{\geq 0} \rightarrow \mathbb{R}$ equipped with the metric d( f, g) = sup{t: f(t) ≠ g(t)} is a universal metric space for all ultrametric spaces (Lemin–Lemin, 1996).

  The universality can be defined also for mappings, other than isometric embeddings, of metric spaces, say, a bi-Lipschitz embedding, etc. For example, any compact metric space is a continuous image of the Cantor set with the natural metric | x − y | inherited from $\mathbb{R}$, and any complete separable metric space is a continuous image of the space of irrational numbers.

• Constructive metric space

  A constructive metric space is a pair (X, d), where X is a set of constructive objects (say, words over an alphabet), and d is an algorithm converting any pair of elements of X into a constructive real number d(x, y) such that d is a metric on X.

• Computable metric space

  Let $\{x_{n}\}_{n\in \mathbb{N}}$ be a sequence of elements from a given Polish (i.e., complete separable) metric space (X, d) such that the set $\{x_{n}: n \in \mathbb{N}\}$ is dense in (X, d). Let $\mathcal{N}(m,n,k)$ be the Cantor tuple function of a triple $(n,m,k) \in \mathbb{N}^{3}$, and let $\{q_{k}\}_{k\in \mathbb{N}}$ be a fixed total standard numbering of the set $\mathbb{Q}$ of rational numbers.

  The triple $(X,d,\{x_{n}\}_{n\in \mathbb{N}})$ is called an effective (or semicomputable) metric space ([Hemm02]) if the set $\{\mathcal{N}(n,m,k): d(x_{m},x_{n}) < q_{k}\}$ is recursively enumerable, i.e., there is an algorithm that enumerates the members of this set. If, moreover, the set $\{\mathcal{N}(n,m,k): d(s_{m},s_{m}) > q_{k}\}$ is recursively enumerable, then this triple is called (Lacombe, 1951) computable metric space, (or recursive metric space ). In other words, the map $d \circ (q \times q): \mathbb{N}^{2} \rightarrow \mathbb{R}$ is a computable (double) sequence of real numbers, i.e., is recursive over $\mathbb{R}$.