Distances on Convex Bodies, Cones, and Simplicial Complexes

Abstract

A convex body in the n-dimensional Euclidean space $\mathbb{E}^{n}$ is a convex compact connected subset of $\mathbb{E}^{n}$. It is called solid (or proper) if it has nonempty interior. Let K denote the space of all convex bodies in $\mathbb{E}^{n}$, and let K _p be the subspace of all proper convex bodies. Given a set $X \subset \mathbb{E}^{n}$, its convex hull c o n v(X) is the minimal convex set containing X.

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1 Distances on Convex Bodies

A convex body in the n-dimensional Euclidean space $\mathbb{E}^{n}$ is a convex compact connected subset of $\mathbb{E}^{n}$. It is called solid (or proper) if it has nonempty interior. Let K denote the space of all convex bodies in $\mathbb{E}^{n}$, and let K _p be the subspace of all proper convex bodies. Given a set $X \subset \mathbb{E}^{n}$, its convex hull c o n v(X) is the minimal convex set containing X.

Any metric space (K, d) on K is called a metric space of convex bodies. Such spaces, in particular the metrization by the Hausdorff metric, or by the symmetric difference metric, play a basic role in Convex Geometry (see, for example, [Grub93]).

For C, D ∈ K∖{∅}, the Minkowski addition and the Minkowski nonnegative scalar multiplication are defined by $C + D =\{ x + y: x \in C,y \in D\}$, and α C = {α x: x ∈ C}, α ≥ 0, respectively. The Abelian semigroup (K, +) equipped with nonnegative scalar multiplication operators can be considered as a convex cone.

The support function $h_{C}: S^{n-1} \rightarrow \mathbb{R}$ of C ∈ K is defined by h _C (u) = sup{〈u, x〉: x ∈ C} for any u ∈ S ⁿ⁻¹, where S ⁿ⁻¹ is the (n − 1)-dimensional unit sphere in $\mathbb{E}^{n}$, and 〈, 〉 is the inner product in $\mathbb{E}^{n}$. The width w _C (u) is $h_{C}(u) + h_{C}(-u) = h_{C-C}(u)$. It is the perpendicular distance between the parallel supporting hyperplanes perpendicular to given direction. The mean width is the average of width over all directions in S ⁿ⁻¹.

• Area deviation

  The area deviation (or template metric ) is a metric on the set K _p in $\mathbb{E}^{2}$ (i.e., on the set of plane convex disks) defined by

  $$\displaystyle{A(C\bigtriangleup D),}$$

  where A(. ) is the area, and △ is the symmetric difference. If C ⊂ D, then it is equal to A(D) − A(C).

• Perimeter deviation

  The perimeter deviation is a metric on K _p in $\mathbb{E}^{2}$ defined by

  $$\displaystyle{2p(conv(C \cup D)) - p(C) - p(D),}$$

  where p(. ) is the perimeter. In the case C ⊂ D, it is equal to p(D) − p(C).

• Mean width metric

  The mean width metric is a metric on K _p in $\mathbb{E}^{2}$ defined by

  $$\displaystyle{v2W(conv(C \cup D)) - W(C) - W(D),}$$

  where W(. ) is the mean width: $W(C) = p(C)/\pi$, and p(. ) is the perimeter.

• Florian metric

  The Florian metric is a metric on K defined by

  $$\displaystyle{\int _{S^{n-1}}\vert h_{C}(u) - h_{D}(u)\vert d\sigma (u) = \vert \vert h_{C} - h_{D}\vert \vert _{1}.}$$

  It can be expressed in the form $2S(conv(C \cup D)) - S(C) - S(D)$ for n = 2 (cf. perimeter deviation ); it can be expressed also in the form $nk_{n}(2W(conv(C \cup D)) - W(C) - W(D))$ for n ≥ 2 (cf. mean width metric ).

  Here S(. ) is the surface area, k _n is the volume of the unit ball $\overline{B}^{n}$ of $\mathbb{E}^{n}$, and W(. ) is the mean width: $W(C) = \frac{1} {nk_{n}}\int _{S^{n-1}}(h_{C}(u) + h_{C}(-u))d\sigma (u)$.

• McClure–Vitale metric

  Given 1 ≤ p ≤ ∞, the McClure–Vitale metric is a metric on K, defined by

  $$\displaystyle{\left (\int _{S^{n-1}}\vert h_{C}(u) - h_{D}(u)\vert ^{p}d\sigma (u)\right )^{\frac{1} {p} } = \vert \vert h_{C} - h_{D}\vert \vert _{p}.}$$

• Pompeiu–Hausdorff–Blaschke metric

  The Pompeiu–Hausdorff–Blaschke metric is a metric on K defined by

  $$\displaystyle{\max \{\sup _{x\in C}\inf _{y\in D}\vert \vert x - y\vert \vert _{2},\sup _{y\in D}\inf _{x\in C}\vert \vert x - y\vert \vert _{2}\},}$$

  where | | . | | ₂ is the Euclidean norm on $\mathbb{E}^{n}$.

  In terms of support functions and using Minkowski addition, this metric is

  $$\displaystyle{\sup _{u\in S^{n-1}}\vert h_{C}(u) - h_{D}(u)\vert = \vert \vert h_{C} - h_{D}\vert \vert _{\infty } =\inf \{\lambda \geq 0: C \subset D +\lambda \overline{B}^{n},D \subset C +\lambda \overline{B}^{n}\},}$$

  where $\overline{B}^{n}$ is the unit ball of $\mathbb{E}^{n}$. This metric can be defined using any norm on $\mathbb{R}^{n}$ and for the space of bounded closed subsets of any metric space.

• Pompeiu–Eggleston metric

  The Pompeiu–Eggleston metric is a metric on K defined by

  $$\displaystyle{\sup _{x\in C}\inf _{y\in D}\vert \vert x - y\vert \vert _{2} +\sup _{y\in D}\inf _{x\in C}\vert \vert x - y\vert \vert _{2},}$$

  where | | . | | ₂ is the Euclidean norm on $\mathbb{E}^{n}$.

  In terms of support functions and using Minkowski addition, this metric is

  $$\displaystyle\begin{array}{rcl} & & \max \{0,\sup _{u\in S^{n-1}}(h_{C}(u) - h_{D}(u))\} +\max \{ 0,\sup _{u\in S^{n-1}}(h_{D}(u) - h_{C}(u))\} = {}\\ & & \quad =\inf \{\lambda \geq 0: C \subset D +\lambda \overline{B}^{n}\}+\inf \{\lambda \geq 0: D \subset C +\lambda \overline{B}^{n}\}, {}\\ \end{array}$$

  where $\overline{B}^{n}$ is the unit ball of $\mathbb{E}^{n}$. This metric can be defined using any norm on $\mathbb{R}^{n}$ and for the space of bounded closed subsets of any metric space.

• Sobolev distance

  The Sobolev distance is a metric on K defined by

  $$\displaystyle{\vert \vert h_{C} - h_{D}\vert \vert _{w},}$$

  where | | . | |  _w is the Sobolev 1-norm on the set $G_{S^{n-1}}$ of all real continuous functions on the unit sphere S ⁿ⁻¹ of $\mathbb{E}^{n}$.

  The Sobolev 1-norm is defined by $\vert \vert \,f\vert \vert _{w} =\langle \, f,f\rangle _{w}^{1/2}$, where 〈, 〉 _w is an inner product on $G_{S^{n-1}}$, given by

  $$\displaystyle{\langle \,f,g\rangle _{w} =\int _{S^{n-1}}(\,fg + \nabla _{s}(\,f,g))dw_{0},\,\,\,w_{0} = \frac{1} {n \cdot k_{n}}w,}$$

  where ∇ _s ( f, g) = 〈grad _s f, grad _s g〉, 〈, 〉 is the inner product in $\mathbb{E}^{n}$, and grad _s is the gradient on S ⁿ⁻¹ (see [ArWe92]).

• Shephard metric

  The Shephard metric is a metric on K _p defined by

  $$\displaystyle{\ln (1 + 2\inf \{\lambda \geq 0: C \subset D +\lambda (D - D),D \subset C +\lambda (C - C)\}).}$$

• Nikodym metric

  The Nikodym metric (or volume of symmetric difference, Dinghas distance ) is a metric on K _p defined by

  $$\displaystyle{V (C\bigtriangleup D) =\int (1_{x\in C} - 1_{x\in D})^{2}dx,}$$

  where V (. ) is the volume (i.e., the Lebesgue n-dimensional measure), and △ is the symmetric difference. For n = 2, one obtains the area deviation.

  Normalized volume of symmetric difference is a variant of Steinhaus distance defined by

  $$\displaystyle{ \frac{V (C\bigtriangleup D)} {V (C \cup D)}.}$$

• Eggleston distance

  The Eggleston distance (or symmetric surface area deviation ) is a distance on K _p defined by

  $$\displaystyle{S(C \cup D) - S(C \cap D),}$$

  where S(. ) is the surface area. It is not a metric.

• Asplund metric

  The Asplund metric is a metric on the space K _p ∕ ≈ of affine-equivalence classes in K _p defined by

  $$\displaystyle{\ln \inf \{\lambda \geq 1:\exists T: \mathbb{E}^{n} \rightarrow \mathbb{E}^{n}\mbox{ affine},\,x \in \mathbb{E}^{n},C \subset T(D) \subset \lambda C + x\}}$$

  for any equivalence classes C ^∗ and D ^∗ with the representatives C and D, respectively.

• Macbeath metric

  The Macbeath metric is a metric on the space K _p ∕ ≈ of affine-equivalence classes in K _p defined by

  $$\displaystyle{\ln \inf \{\vert \det T \cdot P\vert:\exists T,P: \mathbb{E}^{n} \rightarrow \mathbb{E}^{n}\mbox{ regular affine},C \subset T(D),D \subset P(C)\}}$$

  for any equivalence classes C ^∗ and D ^∗ with the representatives C and D, respectively.

  Equivalently, it can be written as lnδ(C, D) + ln δ(D, C), where $\delta (C,D) =\inf _{T}\{\frac{V (T(D))} {V (C)};C \subset T(D)\}$, and T is a regular affine mapping of $\mathbb{E}^{n}$ onto itself.

• Banach–Mazur metric

  The Banach–Mazur metric is a metric on the space K _po ∕ ∼ of the equivalence classes of proper 0-symmetric convex bodies with respect to linear transformations defined by

  $$\displaystyle{\ln \inf \{\lambda \geq 1:\exists T: \mathbb{E}^{n} \rightarrow \mathbb{E}^{n}\mbox{ linear, }C \subset T(D) \subset \lambda C\}}$$

  for any equivalence classes C ^∗ and D ^∗ with the representatives C and D, respectively.

  It is a special case of the Banach–Mazur distance (Chap. 1).

• Separation distance

  The separation distance between two disjoint convex bodies C and D in $\mathbb{E}^{n}$ (in general, between any two disjoint subsets) $\mathbb{E}^{n}$) is (Buckley, 1985) their Euclidean set-set distance inf{ | | x − y | | ₂: x ∈ C, y ∈ D}, while sup{ | | x − y | | ₂: x ∈ C, y ∈ D} is their spanning distance.

• Penetration depth distance

  The penetration depth distance between two interpenetrating convex bodies C and D in $\mathbb{E}^{n}$ (in general, between any two interpenetrating subsets of $\mathbb{E}^{n}$) is (Cameron–Culley, 1986) defined as the minimum translation distance that one body undergoes to make the interiors of C and D disjoint:

  $$\displaystyle{\min \{\vert \vert t\vert \vert _{2}: interior(C + t) \cap D =\emptyset \}.}$$

  Keerthi–Sridharan, 1991, considered | | t | | ₁- and | | t | |  _∞ -analogs of this distance.

  Cf. penetration distance in Chap. 23 and penetration depth in Chap. 24

• Growth distances

  Let C, D ∈ K _p be two compact convex proper bodies. Fix their seed points p _C  ∈ int C and p _D  ∈ int D; usually, they are the centroids of C and D. The growth function g(C, D) is the minimal number λ > 0, such that

  $$\displaystyle{(\{p_{C}\} +\lambda (C\setminus \{p_{C}\})) \cap (\{p_{D}\} +\lambda (D\setminus \{p_{D}\}))\neq \emptyset.}$$

  It is the amount objects must be grown if g(C, D) > 1 (i.e., C ∩ D = ∅), or contracted if g(C, D) > 1 (i.e., int C ∩ int D ≠ ∅) from their internal seed points until their surfaces just touch. The growth separation distance d _S (C, D) and the growth penetration distance d _P (C, D) ([OnGi96]) are defined as

  $$\displaystyle{d_{S}(C,D) =\max \{ 0,r_{CD}(g(C,D) - 1)\}\,\mbox{ and}\,d_{P}(C,D) =\max \{ 0,r_{CD}(1 - g(C,D))\},}$$

  where r _CD is the scaling coefficient (usually, the sum of radii of circumscribing spheres for the sets C∖{p _C } and D∖{p _D }).

  The one-sided growth distance between disjoint C and D (Leven–Sharir, 1987) is

  $$\displaystyle{-1+\min \lambda> 0: (\{p_{C}\} +\lambda \{ (C\setminus \{p_{C}\})) \cap D\neq \emptyset \}.}$$

• Minkowski difference

  The Minkowski difference on the set of all compact subsets, in particular, on the set of all sculptured objects (or free form objects), of $\mathbb{R}^{3}$ is defined by

  $$\displaystyle{A - B =\{ x - y: x \in A,y \in B\}.}$$

  If we consider object B to be free to move with fixed orientation, the Minkowski difference is a set containing all the translations that bring B to intersect with A. The closest point from the Minkowski difference boundary, ∂(A − B), to the origin gives the separation distance between A and B.

  If both objects intersect, the origin is inside of their Minkowski difference, and the obtained distance can be interpreted as a penetration depth distance.

• Demyanov distance

  Given C ∈ K _p and u ∈ S ⁿ⁻¹, denote, if $\vert \{c \in C:\langle u,c\rangle = h_{C}(u)\}\vert = 1$, this unique point by y(u, C) (exposed point of C in direction u).

  The Demyanov difference A ⊖ B of two subsets A, B ∈ K _p is the closure of

  $$\displaystyle{conv(\cup _{T(A)\cap T(B)}\{y(u,A) - y(u,B)\}),}$$

  where $T(C) =\{ u \in S^{n-1}: \vert \{c \in C:\langle u,c\rangle = h_{C}(u)\}\vert = 1\}$.

  The Demyanov distance between two subsets A, B ∈ K _p is defined by

  $$\displaystyle{\vert \vert A \ominus B\vert \vert =\max _{c\in A\ominus B}\vert \vert c\vert \vert _{2}.}$$

  It is shown in [BaFa07] that | | A ⊖ B | | = sup _α  | | St _α (A) − St _α (M) | | ₂, where St _α (C) is a generalized Steiner point and the supremum is over all “sufficiently smooth” probabilistic measures α.

• Maximum polygon distance

  The maximum polygon distance is a distance between two convex polygons $P = (p_{1},\ldots,p_{n})$ and $Q = (q_{1},\ldots,q_{m})$ defined by

  $$\displaystyle{\max _{i,j}\vert \vert p_{i} - q_{j}\vert \vert _{2},\,\,i \in \{ 1,\ldots,n\},\,\,j \in \{ 1,\ldots,m\}.}$$

• Grenander distance

  Let $P = (p_{1},\ldots,p_{n})$ and $Q = (q_{1},\ldots,q_{m})$ be two disjoint convex polygons, and let L(p _i , q _j ), L(p _l , q _m ) be two intersecting critical support lines for P and Q. Then the Grenander distance between P and Q is defined by

  $$\displaystyle{\vert \vert p_{i} - q_{j}\vert \vert _{2} + \vert \vert p_{l} - q_{m}\vert \vert _{2} - \Sigma (p_{i},p_{l}) - \Sigma (g_{j},q_{m}),}$$

  where | | . | | ₂ is the Euclidean norm, and $\Sigma (p_{i},p_{l})$ is the sum of the edges lengths of the polynomial chain $p_{i},\ldots,p_{l}$.

  Here $P = (p_{1},\ldots,p_{n})$ is a convex polygon with the vertices in standard form, i.e., the vertices are specified according to Cartesian coordinates in a clockwise order, and no three consecutive vertices are collinear. A line L is a line of support of P if the interior of P lies completely to one side of L.

  Given two disjoint polygons P and Q, the line L(p _i , q _j ) is a critical support line if it is a line of support for P at p _i , a line of support for Q at q _j , and P and Q lie on opposite sides of L(p _i , q _j ). In general, a chord [a, b] of a convex body C is called its affine diameter if there is a pair of different hyperplanes each containing one of the endpoints a, b and supporting C.

2 Distances on Cones

A convex cone C in a real vector space V is a subset C of V such that C + C ⊂ C, λ C ⊂ C for any λ ≥ 0. A cone C induces a partial order on V by

$$\displaystyle{x\preceq y\mbox{ if and only if }y - x \in C.}$$

The order ⪯ respects the vector structure of V, i.e., if x ⪯ y and z ⪯ u, then $x + z\preceq y + u$, and if x ⪯ y, then λ x ⪯ λ y, $\lambda \in \mathbb{R}$, λ ≥ 0. Elements x, y ∈ V are called comparable and denoted by x ∼ y if there exist positive real numbers α and β such that α y ⪯ x ⪯ β y. Comparability is an equivalence relation; its equivalence classes (which belong to C or to − C) are called parts (or components, constituents).

Given a convex cone C, a subset $S =\{ x \in C: T(x) = 1\}$, where $T: V \rightarrow \mathbb{R}$ is a positive linear functional, is called a cross-section of C. A convex cone C is called almost Archimedean if the closure of its restriction to any 2D subspace is also a cone.

A convex cone C is called pointed if $C \cup (-C) =\{ 0\}$ and solid if int C ≠ ∅.

• Koszul–Vinberg metric

  Given an open pointed convex cone C, let C ^∗ be its dual cone.

  The Koszul–Vinberg metric on C (Vinberg, 1963, and Koszul, 1965) is an affine invariant Riemannian metric defined as the Hessian g = d ² ψ _C , where $\psi _{C}(x) = -\log \int _{C^{{\ast}}}e^{-(\epsilon,x)}d\epsilon$ for any x ∈ C.

  The Hessian of the entropy (Legendre transform of ψ _C (x)) defines a metric on C ^∗, which ([Barb14]) is equivalent to the Fisher–Rao metric (Sect. 7.2). [Barb14] also observed that Fisher–Souriau metric ([Sour70]) generalises Fisher–Rao metric for Lie group thermodynamics and interpreted it as a geometric heat capacity.

• Invariant distances on symmetric cones

  An open convex cone C in an Euclidean space V is said to be homogeneous if its group of linear automorphisms $G =\{ g \in GL(V ): g(C) = C\}$ act transitively on C. If, moreover, $\overline{C}$ is pointed and C is self-dual with respect to the given inner product 〈, 〉, then it is called a symmetric cone. Any symmetric cone is a Cartesian product of such cones of only 5 types: the cones $Sym(n, \mathbb{R})^{+}$, $Her(n, \mathbb{C})^{+}$ (cf. Chap. 12), $Her(n, \mathbb{H})^{+}$ of positive-definite Hermitian matrices with real, complex or quaternion entries, the Lorentz cone (or forward light cone) $\{(t,x_{1},\ldots,x_{n}) \in \mathbb{R}^{n+1}: t^{2}> x_{1}^{2} +\ldots +x_{n}^{2}\}$ and 27-dimensional exceptional cone of 3 × 3 positive-definite matrices over the octonions $\mathbb{O}$. An n × n quaternion matrix A can be seen as a 2n × 2n complex matrix A′; so, $A \in Her(n, \mathbb{H})^{+}$ means $A' \in Her(2n, \mathbb{C})^{+}$.

  Let V be an Euclidean Jordan algebra, i.e., a finite-dimensional Jordan algebra (commutative algebra satisfying x ²(xy) = x(x ² y) and having a multiplicative identity e) equipped with an associative (〈xy, z〉 = 〈y, xz〉) inner product 〈, 〉. Then the set of square elements of V is a symmetric cone, and every symmetric cone arises in this way. Denote $P(x)y = 2x(xy) - x^{2}y$ for any x, y ∈ C.

  For example, for $C = PD_{n}(\mathbb{R})$, the group G is $GL(n, \mathbb{R})$, the inner product is 〈X, Y 〉 = Tr(XY ), the Jordan product is $\frac{1} {2}(XY + Y X)$, and P(X)Y = XYX, where the multiplication on the right-hand side is the usual matrix multiplication.

  If r is the rank of V, then for any x ∈ V there is a complete set of orthogonal primitive idempotents $c_{1},\ldots,c_{r}\neq 0$ (i.e., c _i ² = c _i , c _i indecomposable, c _i c _j  = 0 if i ≠ j, $\sum _{i=1}^{r}c_{i} = e$) and real numbers $\lambda _{1},\ldots,\lambda _{r}$, called eigenvalues of x, such that $x =\sum _{ i=1}^{r}\lambda _{i}c_{i}$. Let x, y ∈ C and $\lambda _{1},\ldots,\lambda _{r}$ be the eigenvalues of $P(x^{-\frac{1} {2} })y$. Lim, 2001, defined following three G-invariant distances on any symmetric cone C:

  $$\displaystyle{d_{R} = (\sum _{1\leq i\leq r}\ln ^{2}\lambda _{ i})^{\frac{1} {2} },\,\,d_{F} =\max _{1\leq i\leq r}\ln \vert \lambda _{i}\vert,\,\,d_{H} =\ln (\max _{1\leq i\leq r}\lambda _{i}(\min _{1\leq i\leq r}\lambda _{i})^{-1}).}$$

  For above distances, the geometric mean $P(x^{\frac{1} {2} })(P(x^{-\frac{1} {2} }y))^{\frac{1} {2} }$ is the midpoint of x and y. The distances d _R (x, y), d _F (x, y) are the intrinsic metrics of G-invariant Riemannian and Finsler metrics on C. The Riemannian geodesic curve $\alpha (t) = P(x^{\frac{1} {2} })(P(x^{-\frac{1} {2} }y))^{t}$ is one of infinitely many shortest Finsler curves passing through x and y. The space (C, d _R (x, y)) is an Hadamard space (Chap. 6), while (C, d _F (x, y)) is not. The distance d _F (x, y) is the Thompson’s part metric on C, and the distance d _H (x, y) is the Hilbert projective semimetric on C which is a complete metric on the unit sphere on C.

• Thompson’s part metric

  Given a convex cone C in a real Banach space V, the Thompson’s part metric on a part K ⊂ C∖{0} is defined (Thompson, 1963) by

  $$\displaystyle{\log \max \{m(x,y),m(y,x)\}}$$

  for any x, y ∈ K, where $m(x,y) =\inf \{\lambda \in \mathbb{R}: y\preceq \lambda x\}$.

  If C is almost Archimedean, then K equipped with this metric is a complete metric space. If C is finite-dimensional, then one obtains a chord space (Chap. 6). The positive cone $\mathbb{R}_{+}^{n} =\{ (x_{1},\ldots,x_{n}): x_{i} \geq 0\mbox{ for }1 \leq i \leq n\}$ equipped with this metric is isometric to a normed space which can be seen as being flat. The same holds for the Hilbert projective semimetric on $\mathbb{R}_{+}^{n}$.

  If C is a closed solid cone in $\mathbb{R}^{n}$, then int C can be seen as an n-dimensional manifold M ⁿ. If for any tangent vector v ∈ T _p (M ⁿ), p ∈ M ⁿ, we define a norm $\vert \vert v\vert \vert _{p}^{T} =\inf \{\alpha> 0: -\alpha p\preceq v\preceq \alpha p\}$, then the length of any piecewise differentiable curve γ: [0, 1] → M ⁿ is l(γ) = ∫ ₀ ¹ | | γ ^′(t) | |  _γ(t) ^T dt, and the distance between x and y is inf _γ l(γ), where the infimum is taken over all such curves γ with γ(0) = x, γ(1) = y.

• Hilbert projective semimetric

  Given a pointed closed convex cone C in a real Banach space V, the Hilbert projective semimetric on C∖{0} is defined (Bushell, 1973), for x, y ∈ C∖{0}, by

  $$\displaystyle{h(x,y) =\log (m(x,y)m(y,x)),}$$

  where $m(x,y) =\inf \{\lambda \in \mathbb{R}: y\preceq \lambda x\}$; it holds $\frac{1} {m(y,x)} =\sup \{\lambda \in \mathbb{R}:\lambda y\preceq x\}$. This semimetric is finite on the interior of C and h(λ x, λ′y) = h(x, y) for λ, λ′ > 0.

  So, h(x, y) is a metric on the projectivization of C, i.e., the space of rays of this cone.

  If C is finite-dimensional, and S is a cross-section of C (in particular, $S =\{ x \in C: \vert \vert x\vert \vert = 1\}$, where | | . | | is a norm on V ), then, for any distinct points x, y ∈ S, it holds h(x, y) = | ln(x, y, z, t) | , where z, t are the points of the intersection of the line l _x, y with the boundary of S, and (x, y, z, t) is the cross-ratio of x, y, z, t. Cf. the Hilbert projective metric in Chap. 6.

  If C is finite-dimensional and almost Archimedean, then each part of C is a chord space (Chap. 6) under the Hilbert projective semimetric. On the Lorentz cone $L =\{ x = (t,x_{1},\ldots,x_{n}) \in \mathbb{R}^{n+1}: t^{2}> x_{1}^{2} +\ldots +x_{n}^{2}\}$, this semimetric is isometric to the n-dimensional hyperbolic space. On the hyperbolic subspace $H =\{ x \in L:\det (x) = 1\}$, it holds h(x, y) = 2d(x, y), where d(x, y) is the Thompson’s part metric which is (on H) the usual hyperbolic distance arccosh〈x, y〉.

  If C is a closed solid cone in $\mathbb{R}^{n}$, then int C can be seen as an n-manifold M ⁿ (Chap. 2). If for any tangent vector v ∈ T _p (M ⁿ), p ∈ M ⁿ, we define a seminorm $\vert \vert v\vert \vert _{p}^{H} = m(p,v) - m(v,p)$, then the length of any piecewise differentiable curve γ: [0, 1] → M ⁿ is l(γ) = ∫ ₀ ¹ | | γ ^′(t) | |  _γ(t) ^H dt, and h(x, y) = inf _γ l(γ), where the infimum is taken over all such curves γ with γ(0) = x and γ(1) = y.

• Bushell metric

  Given a convex cone C in a real Banach space V, the Bushell metric on the set $S =\{ x \in C:\sum _{ i=1}^{n}\vert x_{i}\vert = 1\}$ (in general, on any cross-section of C) is defined by

  $$\displaystyle{\frac{1 - m(x,y) \cdot m(y,x)} {1 + m(x,y) \cdot m(y,x)}}$$

  for any x, y ∈ S, where $m(x,y) =\inf \{\lambda \in \mathbb{R}: y\preceq \lambda x\}$. In fact, it is equal to $\tanh (\frac{1} {2}h(x,y))$, where h is the Hilbert projective semimetric.

• k-oriented distance

  A simplicial cone C in $\mathbb{R}^{n}$ is defined as the intersection of n (open or closed) half-spaces, each of whose supporting planes contain the origin 0. For any set M of n points on the unit sphere, there is a unique simplicial cone C that contains these points. The axes of the cone C can be constructed as the set of the n rays, where each ray originates at the origin, and contains one of the points from M.

  Given a partition $\{C_{1},\ldots,C_{k}\}$ of $\mathbb{R}^{n}$ into a set of simplicial cones C ₁, $\ldots$, C _k , the k-oriented distance is a metric on $\mathbb{R}^{n}$ defined by

  $$\displaystyle{d_{k}(x - y)}$$

  for all $x,y \in \mathbb{R}^{n}$, where, for any x ∈ C _i , the value d _k (x) is the length of the shortest path from the origin 0 to x traveling only in directions parallel to the axes of C _i .

• Cones over metric space

  A cone over a metric space (X, d) is the quotient space Con(X, d)=(X × [0, 1])∕(X ×{ 0}) obtained from the product $X \times \mathbb{R}_{\geq 0}$ by collapsing the fiber (subspace X ×{ 0}) to a point (the apex of the cone). Cf. metric cone in Chap. 1.

  The Euclidean cone over the metric space (X, d) is the cone Con(X, d) with a metric d defined, for any (x, t), (y, s) ∈ Con(X, d), by

  $$\displaystyle{\sqrt{t^{2 } + s^{2 } - 2ts\cos (\min \{d(x, y),\pi \} )}.}$$

  If (X, d) is a compact metric space with diameter < 2, the Krakus metric is a metric on Con(X, d) defined, for any (x, t), (y, s) ∈ Con(X, d), by

  $$\displaystyle{\min \{s,t\}d(x,y) + \vert t - s\vert.}$$

  The cone Con(X, d) with the Krakus metric admits a unique midpoint for each pair of its points if (X, d) has this property.

  If M ⁿ is a manifold with a pseudo-Riemannian metric g, one can consider a metric dr ² + r ² g (in general, a metric $\frac{1} {k}dr^{2} + r^{2}g$, k ≠ 0) on $Con(M^{n}) = M^{n} \times \mathbb{R}_{>0}$. For example, $Con(M^{n}) = \mathbb{R}^{n}\setminus \{0\}$ if (M ⁿ, g) is the unit sphere in $\mathbb{R}^{n}$.

  A spherical cone (or suspension) $\Sigma (X)$ over a metric space (X, d) is the quotient of the product X × [0, a] obtained by identifying all points in the fibers X ×{ 0} and X ×{ a}. If (X, d) is a length space (Chap. 6) with diam(X) ≤ π, and a = π, the suspension metric on $\Sigma (X)$ is defined, for any $(x,t),(y,s) \in \Sigma (X)$, by

  $$\displaystyle{\arccos (\cos t\cos s +\sin t\sin s\cos d(x,y)).}$$

3 Distances on Simplicial Complexes

An r-dimensional simplex (or geometrical simplex, hypertetrahedron) is the convex hull of r + 1 points of $\mathbb{E}^{n}$ which do not lie in any (r − 1)-plane. The boundary of an r-simplex has r + 1 0-faces (polytope vertices), $\frac{r(r+1)} {2}$ 1-faces (polytope edges), and $(_{i+1}^{r+1})$ i-faces, where ( _i ^r) is the binomial coefficient. The content (i.e., the hypervolume) of a simplex can be computed using the Cayley–Menger determinant. The regular simplex of dimension r is denoted by α _r . Simplicial depth of a point $p \in \mathbb{E}^{n}$ relative to a set $P \subset \mathbb{E}^{n}$ is the number of simplices S, generated by (n + 1)-subsets of P and containing p.

Roughly, a geometrical simplicial complex is a space with a triangulation, i.e., a decomposition of it into closed simplices such that any two simplices either do not intersect or intersect only along a common face.

An abstract simplicial complex S is a set, whose elements are called vertices, in which a family of finite nonempty subsets, called simplices, is distinguished, such that every nonempty subset of a simplex s is a simplex, called a face of s, and every one-element subset is a simplex. A simplex is called i-dimensional if it consists of i + 1 vertices. The dimension of S is the maximal dimension of its simplices. For every simplicial complex S there exists a triangulation of a polyhedron whose simplicial complex is S. This geometric simplicial complex, denoted by GS, is called the geometric realization of S.

• Vietoris–Rips complex

  Given a metric space (X, d) and distance δ, their Vietoris–Rips complex is an abstract simplicial complex, the simplexes of which are the finite subsets M of (X, d) having diameter at most δ; the dimension of a simplex defined by M is | M | − 1.

• Simplicial metric

  Given an abstract simplicial complex S, the points of geometric simplicial complex GS, realizing S, can be identified with the functions α: S → [0, 1] for which the set {x ∈ S: α(x) ≠ 0} is a simplex in S, and ∑ _x ∈ S α(x) = 1. The number α(x) is called the x-th barycentric coordinate of α.

  The simplicial metric on GS (Lefschetz, 1939) is the Euclidean metric on it:

  $$\displaystyle{\sqrt{\sum _{x\in S } (\alpha (x) -\beta (x))^{2}}.}$$

  Tukey, 1939, found another metric on GS, topologically equivalent to a simplicial one. His polyhedral metric is the intrinsic metric, defined as the infimum of the lengths of the polygonal lines joining the points α and β such that each link is within one of the simplices. An example of a polyhedral metric is the intrinsic metric on the surface of a convex polyhedron in $\mathbb{E}^{3}$.

• Polyhedral space

  A Euclidean polyhedral space is a simplicial complex with a polyhedral metric. Every simplex is a flat space (a metric space locally isometric to some $\mathbb{E}^{n}$; cf. Chap. 1), and the metrics of any two simplices coincide on their intersection. The metric is the maximal metric not exceeding the metrics of simplices.

  If such a space is an n-manifold (Chap. 2), a point in it is a metric singularity if it has no neighborhood isometric to an open subset of $\mathbb{E}^{n}$.

  A polyhedral metric on a simplicial complex in a space of constant (positive or negative) curvature results in spherical and hyperbolic polyhedral spaces.

  The dimension of a polyhedral space is the maximal dimension of simplices used to glue it. Metric graphs (Chap. 15) are just one-dimensional polyhedral spaces.

  The surface of a convex polyhedron is a 2D polyhedral space. A polyhedral metric d on a triangulated surface is a circle-packing metric (Thurston, 1985) if there exists a vertex-weighting w(x) > 0 with $d(x,y) = w(x) + w(y)$ for any edge xy.

• Manifold edge-distance

  A (boundaryless) combinatorial n-manifold is an abstract n-dimensional simplicial complex M ⁿ in which the link of each r-simplex is an $(n - r - 1)$-sphere. The category of such spaces is equivalent to the category of piecewise-linear (PL) manifolds.

  The link of a simplex S is Cl(Star _S ) − Star _S , where Star _S is the set of all simplices in M ⁿ having a face S, and Cl(Star _S ) is the smallest simplicial subcomplex of M ⁿ containing Star _S .

  The edge-distance between vertices u, v ∈ M ⁿ is the minimum number of edges needed to connect them.

• Manifold triangulation metric

  Let M ⁿ be a compact PL (piecewise-linear) n-dimensional manifold. A triangulation of M ⁿ is a simplicial complex such that its corresponding polyhedron is PL-homeomorphic to M ⁿ. Let $T_{M^{n}}$ be the set of all combinatorial types of triangulations, where two triangulations are equivalent if they are simplicially isomorphic.

  Every such triangulation can be seen as a metric on the smooth manifold M if one assigns the unit length for any of its 1-dimensional simplices; so, $T_{M^{n}}$ can be seen as a discrete analog of the space of Riemannian structures, i.e., isometry classes of Riemannian metrics on M ⁿ.

  A manifold triangulation metric between two triangulations x and y is (Nabutovsky and Ben-Av, 1993) an editing metric on $T_{M^{n}}$, i.e., the minimal number of elementary moves, from a given finite list of operations, needed to obtain y from x.

  For example, the bistellar move consists of replacing a subcomplex of a given triangulation, which is simplicially isomorphic to a subcomplex of the boundary of the standard (n + 1)-simplex, by the complementary subcomplex of the boundary of an (n + 1)-simplex, containing all remaining n-simplices and their faces. Every triangulation can be obtained from any other one by a finite sequence of bistellar moves.

• Polyhedral chain metric

  An r-dimensional polyhedral chain A in $\mathbb{E}^{n}$ is a linear expression ∑ _i = 1 ^m d _i t _i ^r, where, for any i, the value t _i ^r is an r-dimensional simplex of $\mathbb{E}^{n}$. The boundary ∂ A of a chain AD is the linear combination of boundaries of the simplices in the chain. The boundary of an r-dimensional chain is an (r − 1)-dimensional chain.

  A polyhedral chain metric is a norm metric | | A − B | | on the set $C_{r}(\mathbb{E}^{n})$ of all r-dimensional polyhedral chains. As a norm | | . | | on $C_{r}(\mathbb{E}^{n})$ one can take:

  • 1. The mass of a polyhedral chain, i.e., | A | = ∑ _i = 1 ^m | d _i  | | t _i ^r | , where | t ^r | is the volume of the cell t _i ^r;

  • 2. The flat norm of a polyhedral chain, i.e., | A | ^♭ = inf _D { | A − ∂ D | + | D | }, where the infimum is taken over all (r + 1)-dimensional polyhedral chains;

  • 3. The sharp norm of a polyhedral chain, i.e.,

    $$\displaystyle{\vert A\vert ^{\sharp } =\inf \left (\frac{\sum _{i=1}^{m}\vert d_{ i}\vert \vert t_{i}^{r}\vert \vert v_{ i}\vert } {r + 1} + \vert \sum _{i=1}^{m}d_{ i}T_{v_{i}}t_{i}^{r}\vert ^{\flat }\right ),}$$

    where the infimum is taken over all shifts v (here T _v t ^r is the cell obtained by shifting t ^r by a vector v of length | v | ). A flat chain of finite mass is a sharp chain. If r = 0, than | A | ^♭ = | A | ^♯.

  The metric space of polyhedral co-chains (i.e., linear functions of polyhedral chains) can be defined similarly. As a norm of a polyhedral co-chain X one can take:

  • 1. The co-mass of a polyhedral co-chain, i.e., $\vert X\vert =\sup _{\vert A\vert =1}\vert X(A)\vert$, where X(A) is the value of the co-chain X on a chain A;

  • 2. The flat co-norm of a polyhedral co-chain, i.e., $\vert X\vert ^{\flat } =\sup _{\vert A\vert ^{\flat }=1}\vert X(A)\vert$;

  • 3. The sharp co-norm of a polyhedral co-chain, i.e., $\vert X\vert ^{\sharp } =\sup _{\vert A\vert ^{\sharp }=1}\vert X(A)\vert$.