Distances Defined by Zonoids

Abstract

The goals of this chapter are to: • Introduce $\mathfrak{N}$-distances defined by zonoids, • Explain the connections between $\mathfrak{N}$-distances and zonoids.

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.

The goals of this chapter are to:

• Introduce $\mathfrak{N}$-distances defined by zonoids,

• Explain the connections between $\mathfrak{N}$-distances and zonoids.

Notation introduced in this chapter:

Notation	Description
h(K, u)	Support function of a convex body
K 1 ⊕ K 2	Minkowski sum of sets K 1 and K 2
S d − 1	Unit sphere in ℝd

1 Introduction

Suppose that $\mathfrak{X}$ is a metric space with the distance ρ. It is well known (Schoenberg 1938) that $\mathfrak{X}$ is isometric to a subspace of a Hilbert space if and only if ρ² is a negative definite kernel. The so-called $\mathfrak{N}$-distance (Klebanov 2005) is a variant of a construction of a distance on a space of measures on $\mathfrak{X}$ such that ${\mathfrak{N}}^{2}$ is a negative definite kernel. Such a construction is possible if and only if ρ² is a strongly negative definite kernel on $\mathfrak{X}$.

In this chapter, we show that the supporting function of any zonoid in ℝ^d is a negative definite first-degree homogeneous function. The inverse is also true. If the support of a generating measure of a zonoid coincides with the unit sphere, then the supporting function is strongly negative definite, and therefore it generates a distance on the space of Borel probability measures on ℝ^d.

2 Main Notions and Definitions

Here we review some known definitions and facts from stochastic geometry.¹

Let $\mathfrak{C}$ (resp. ${\mathfrak{C}}^{{\prime}}$) be the system of all compact convex sets (resp. nonempty compact convex sets) in ℝ^d. A set $K \in {\mathfrak{C}}^{{\prime}}$ is called a convex body if $K \in {\mathfrak{C}}^{{\prime}}$; then for each u ∈ S ^{d − 1} there is exactly one number h(K, u) such that the hyperplane

$$\{x \in {\mathbb{R}}^{d} :\langle x,u\rangle - h(K,u) = 0\}$$

(25.2.1)

intersects K, and ⟨x, u⟩ − h(K, u) ≤ 0 for each x ∈ K. This hyperplane is called the support hyperplane, and the function h(K, u), u ∈ S ^{d − 1} (where S ^{d − 1} is the unit sphere), is the support function (restricted to S ^{d − 1}) of K. Equivalently, one can define

$$h(K,u) =\sup \{\langle x,u\rangle, \;x \in K\},\;\;u \in {\mathbb{R}}^{d}.$$

(25.2.2)

Its geometrical meaning is the signed distance of the support hyperplane from the coordinate origin.

An important property of h(K, u) is its additivity:

$$h(K_{1} \oplus K_{2},u) = h(K_{1},u) + h(K_{2},u),$$

where $K_{1} \oplus K_{2} =\{ a + b :\; a \in K_{1},\,b \in K_{2}\}$ is the Minkowski sum of K ₁ and K ₂. For $K \in {\mathfrak{C}}^{{\prime}}$ let Ǩ$=\{ -k,k \in K\}$. We say that K is centrally symmetric if K ^′ = Ǩ ^′ for some translate K ^′, i.e., if K has a center of symmetry.

The Minkowski sum of finitely many centered line segments is called a zonotope. Consider a zonotope

$$\mathcal{Z} = \bigoplus\limits_{i=1}^{k}a_{ i}[v_{i},-v_{i}],$$

(25.2.3)

where a _i  > 0, $v_{i} \in {\mathbb{S}}^{d-1}$. Its support function is given by

$$h(\mathcal{Z},u) = h_{\mathcal{Z}}(u) = \sum\limits_{i=1}^{k}a_{ i}\vert \langle u,v_{i}\rangle \vert.$$

(25.2.4)

We use the notation ${\mathcal{K}}^{{\prime}}$ for the space of all compact subsets of ℝ^d with the Hausdorff metric

$$d_{H}(K_{1},K_{2}) =\max \{\mathop{\sup}\limits_{x\in K_{1}}\mathrm{dist}(x,K_{2}),\mathop{\sup}\limits_{y\in K_{2}}\mathrm{dist}(y,K_{1})\},$$

(25.2.5)

where $\mathrm{dist}(x,K) =\mathop{\inf}_{z\in K}\|x - z\|$.

A set $\mathcal{Z}\in {\mathfrak{C}}^{{\prime}}$ is called a zonoid if it is a limit in a d _H distance of a sequence of zonotopes.

It is known that a convex body $\mathcal{Z}$ is a zonoid if and only if its support function has a representation

$$h(\mathcal{Z},u) = \int\nolimits_{{\mathbb{S}}^{d-1}}^{}\vert \langle u,v\rangle \vert \mathrm{d}\mu _{\mathcal{Z}}(v)$$

(25.2.6)

for an even measure $\mu _{\mathcal{Z}}$ on ${\mathbb{S}}^{d-1}$. The measure $\mu _{\mathcal{Z}}$ is called the generating measure of $\mathcal{Z}$. It is known that the generating measure is unique for each zonoid $\mathcal{Z}$.

3 $\mathfrak{N}$-Distances

Suppose that $(\mathfrak{X},\mathfrak{A})$ is a measurable space and $\mathcal{L}$ is a strongly negative definite kernel on $\mathfrak{X}$. Denote by $\mathcal{B}_{\mathcal{L}}$ the set of all probabilities μ on $(\mathfrak{X},\mathfrak{A})$ for which there exists the integral

$$\int\nolimits_{\mathfrak{X}}^{} \int\nolimits_{\mathfrak{X}}^{}\mathcal{L}(x,y)\mathrm{d}\mu (x)\mathrm{d}\mu (y) < \infty.$$

(25.3.1)

For $\mu, \nu \in \mathcal{B}_{\mathcal{L}}$ consider

$$\begin{array}{rcl} \mathcal{N}(\mu, \nu )& =& 2\int\nolimits_{\mathfrak{X}}^{} \int\nolimits_{\mathfrak{X}}^{}\mathcal{L}(x,y)\mathrm{d}\mu (x)\mathrm{d}\nu (y) \\ & & -\int\nolimits_{\mathfrak{X}}^{} \int\nolimits_{\mathfrak{X}}^{}\mathcal{L}(x,y)\mathrm{d}\mu (x)\mathrm{d}\mu (y) \\ & & -\int\nolimits_{\mathfrak{X}}^{} \int\nolimits_{\mathfrak{X}}^{}\mathcal{L}(x,y)\mathrm{d}\nu (x)\mathrm{d}\nu (y).\end{array}$$

(25.3.2)

It is known (Klebanov 2005) that

$$\mathfrak{N}(\mu, \nu ) ={\Bigl ( \mathcal{N}{(\mu, \nu )\Bigr )}}^{1/2}$$

is a distance on $\mathcal{B}_{\mathcal{L}}$.

Described below are some examples of negative definite kernels.

Example 25.3.1.

Let $\mathfrak{X} = {\mathbb{R}}^{1}$. For r ∈ [0, 2] define

$$\mathcal{L}_{r}(x,y) = \vert x - y{\vert }^{r}.$$

The function $\mathcal{L}_{r}$ is a negative definite kernel. For r ∈ (0, 2), $\mathcal{L}_{r}$ is a strongly negative definite kernel.

For the proof of the statement in this example and the statement in the next example (Example 25.3.2), See Klebanov [2005].

Example 25.3.2.

Let $\mathcal{L}(x,y) = f(x - y)$, where f(t) is a continuous function on ℝ^d, f(0) = 0, $f(-t) = f(t)$. $\mathcal{L}$ is a negative definite kernel if and only of

$$f(t) = \int\nolimits_{{\mathbb{R}}^{d}}^{}{\bigl (1 -\cos \langle t,u\rangle \bigr )}\frac{1 +\| {u\|}^{2}} {\|{u\|}^{2}} \mathrm{d}\Theta (u),$$

(25.3.3)

where Θ is a finite measure on ℝ^d. Representation (25.3.3) is unique. Kernel $\mathcal{L}$ is strongly negative definite if the support of the measure Θ coincides with the whole space ℝ^d.

We will give an alternative proof for the fact that | x − y | is a negative definite kernel. For the case $\mathfrak{X} = {\mathbb{R}}^{1}$ define

$$\mathcal{L}(x,y) = 2\max (x,y) - x - y = \vert x - y\vert.$$

(25.3.4)

Then $\mathcal{L}$ is a negative definite kernel.

Proof.

It is sufficient to show that max(x, y) is a negative definite kernel. For arbitrary a ∈ ℝ¹ consider

$$u_{a}(x) = \left \{\begin{array}{@{}l@{\quad }l@{}} 1,\quad &x < a,\\ 0,\quad &x \geq a. \end{array} \right.$$

(25.3.5)

It is clear that

$$u_{a}(\max (x,y)) = u_{a}(x)u_{a}(y).$$

Let F(a) be a nondecreasing bounded function on ℝ¹. Define

$$\mathcal{K}(x,y) = \int\nolimits_{-\infty }^{\infty }u_{ a}(\max (x,y))\mathrm{d}F(a).$$

For any integer n > 1 and arbitrary c ₁, …, c _n under condition $\sum_{j=1}^{n}c_{j} = 0$ we have

$$\begin{array}{rcl} \sum\limits_{i=1}^{n} \sum\limits_{j=1}^{n}\mathcal{K}(x_{ i},x_{j})c_{i}c_{j}& =& \int\nolimits_{-\infty }^{\infty }\sum\limits_{i=1}^{n} \sum\limits_{j=1}^{n}u_{ a}(x_{i})u_{a}(x_{j})c_{i}c_{j}\mathrm{d}F(a) \\ & =& \int\nolimits_{-\infty }^{\infty }{\left (\sum\limits_{i=1}^{n}u_{ a}(x_{i})c_{i}\right )}^{2}\mathrm{d}F(a) \geq 0. \\ & & \\ \end{array}$$

But

$$\begin{array}{rcl} \mathcal{K}(x,y)& =& \int\nolimits_{-\infty }^{\infty }u_{ a}(\max (x,y))\mathrm{d}F(a) \\ & =& F(+\infty ) - F(\max (x,y)). \\ & & \\ \end{array}$$

Let us fix arbitrary A > 0 and apply the previous equality to the function

$$F(a) = F_{A}(a) = \left \{\begin{array}{@{}l@{\quad }l@{}} A \quad &\text{ for}\;a > A, \\ a \quad &\text{ for}\; - A \leq a \leq A, \\ -A\quad &\text{ for}\;a < -A. \end{array} \right.$$

(25.3.6)

In this case, $\mathcal{K}(x,y) = A -\max (x,y)$ for x, y ∈ [ − A, A], and, as A → ∞, we obtain that max(x, y) is a negative definite kernel.

Directly from the definition of a negative definite kernel and Example 25.3.1 we obtain the next example.

Example 25.3.3.

Let x, y ∈ ℝ^d, and f :   ℝ^d → ℝ¹. Define

$$\mathcal{L}(x,y) = \vert f(x) - f(y)\vert.$$

Then $\mathcal{L}$ is a negative definite kernel.

Of course, the mixture of negative definite kernels is again a negative definite kernel.

Example 25.3.4.

Let us choose and fix a vector $\theta \in {\mathbb{S}}^{d-1}$ and consider the kernel

$$\mathcal{L}_{\theta }(x,y) = \vert \langle x,\theta \rangle -\langle y,\theta \rangle \vert.$$

From previous considerations it is clear that $\mathcal{L}_{\theta }$ is a negative definite kernel on ℝ^d, and for the σ-finite measure Ξ

$$\mathcal{L}_{\Xi }(x,y) = \int\nolimits_{{\mathbb{S}}^{d-1}}^{}\mathcal{L}_{\theta }(x,y)\mathrm{d}\Xi (\theta )$$

(25.3.7)

is, again, a negative definite kernel.

Consider expression (25.3.2) constructed on the basis of (25.3.7). Let us rewrite (25.3.2) in a different form. Suppose that X and Y are two random vectors in ℝ^d with distributions μ and ν, respectively. We write $\mathcal{N}(X,Y )$ instead of $\mathcal{N}(\mu, \nu )$, so that

$$\mathcal{N}(X,Y ) = 2E\mathcal{L}_{\Xi }(X,Y ) - E\mathcal{L}_{\Xi }(X,{X}^{{\prime}}) - E\mathcal{L}_{ \Xi }(Y,{Y }^{{\prime}}),$$

where ${X}^{{\prime}}\stackrel{d}{=}X$ and ${Y }^{{\prime}}\stackrel{d}{=}Y$ are independent copies of X and Y, respectively. Note that we use the sign $\stackrel{d}{=}$ for the equality in a distribution. We have

$$\begin{array}{rcl} \mathcal{N}(X,Y )& =& E\int\nolimits_{{\mathbb{S}}^{d-1}}^{}[4\max (\langle X,\theta \rangle, \langle Y,\theta \rangle ) \\ & & -2\max (\langle X,\theta \rangle, \langle {X}^{{\prime}},\theta \rangle ) - 2\max (\langle Y,\theta \rangle, \langle {Y }^{{\prime}},\theta \rangle )]\mathrm{d}\Xi (\theta ).\end{array}$$

Denote X _θ = ⟨X, θ⟩, Y _θ = ⟨Y, θ⟩. Then

$$\begin{array}{rcl} \mathcal{N}(X,Y )& =& 2\int\nolimits_{{\mathbb{S}}^{d-1}}^{}\lim\limits _{A\rightarrow \infty }E\int\nolimits_{-A}^{A}\bigl (u_{ a}(X_{\theta })u_{a}(X_{\theta }^{{\prime}}) \\ & & +u_{a}(Y _{\theta })u_{a}(Y _{\theta }^{{\prime}}) - 2u_{ a}(X_{\theta })u_{a}(Y _{\theta })\bigr )\mathrm{d}F_{A}(a)\mathrm{d}\Xi (\theta )\end{array}$$

But Eu _a (X _θ) = Pr{X _θ < a}, and therefore

$$\begin{array}{rcl} \mathcal{N}(X,Y )& =& 2\lim\limits _{A\rightarrow \infty }\int\nolimits_{{\mathbb{S}}^{d-1}}^{}\mathrm{d}\Xi (\theta )\int\nolimits_{-A}^{A}\Bigl (\mathrm{Pr}\{X_{ \theta } < a\}\mathrm{Pr}\{X_{\theta }^{{\prime}} < a\} \\ & & +\mathrm{Pr}\{Y _{\theta } < a\}\mathrm{Pr}\{Y _{\theta }^{{\prime}} < a\} - 2\mathrm{Pr}\{X_{ \theta } < a\}\mathrm{Pr}\{Y _{\theta } < a\}\Bigr )\mathrm{d}F_{A}(a) \\ & =& 2\int\nolimits_{{\mathbb{S}}^{d-1}}^{}\mathrm{d}\Xi (\theta )\int\nolimits_{-\infty }^{\infty }{\Bigl (F_{ \theta }(a) - G_{\theta }{(a)\Bigr )}}^{2}\mathrm{d}a, \\ \end{array}$$

where F _θ(a) = Pr{X _θ < a}, G _θ(a) = Pr{Y _θ < a}. So finally we have

$$\mathcal{N}(X,Y ) = 2\int\nolimits_{{\mathbb{S}}^{d-1}}^{}\mathrm{d}\Xi (\theta )\int\nolimits_{-\infty }^{\infty }{\Bigl (F_{ \theta }(a) - G_{\theta }{(a)\Bigr )}}^{2}\mathrm{d}a.$$

(25.3.8)

If the support of Ξ coincides with ${\mathbb{S}}^{d-1}$, then $\mathfrak{N}(X,Y ) ={\Bigl ( \mathcal{N}{(X,Y )\Bigr )}}^{1/2}$ is a distance between the distributions of X and Y.

Let us return to the kernel

$$\mathcal{L}_{\theta }(x,y) = 2\max (\langle x,\theta \rangle, \langle y,\theta \rangle ) -\langle x,\theta \rangle -\langle y,\theta \rangle.$$

Choose arbitrary $\theta _{o} \in {\mathbb{S}}^{d-1}$, and consider the measure

$$\Xi _{o} = \frac{1} {2}{\bigl (\delta _{\theta _{o}} + \delta _{-\theta _{o}}\bigr )},$$

where $\delta _{\theta _{o}}$ is the measure concentrated at point θ _o . Then

$$\begin{array}{rcl} \mathcal{L}_{\Xi _{\theta _{o}}}(x,y)& =& \int\nolimits_{{\mathbb{S}}^{d-1}}^{}\mathcal{L}_{\theta }(x,y)\mathrm{d}\Xi _{o}(\theta ) \\ & =& \max (\langle x,\theta _{o}\rangle, \langle y,\theta _{o}\rangle ) +\max (-\langle x,\theta _{o}\rangle, -\langle y,\theta _{o}\rangle ) \\ & =& \vert \langle x - y,\theta \rangle \vert. \end{array}$$

Now, if we have an arbitrary even measure Ξ _s on sphere ${\mathbb{S}}^{d-1}$, then

$$\begin{array}{rcl} \mathcal{L}_{\Xi _{s}}(x,y)& =& \int\nolimits_{{\mathbb{S}}^{d-1}}^{}\mathcal{L}_{\theta }(x,y)\mathrm{d}\Xi _{s}(\theta ) \\ & =& \int\nolimits_{{\mathbb{S}}^{d-1}}^{}\vert \langle x - y,\theta \rangle \vert \mathrm{d}\Xi _{s}(\theta ) \\ \end{array}$$

is a negative definite kernel. Let us note that the function

$$h(z) = \int\nolimits_{{\mathbb{S}}^{d-1}}^{}\vert \langle z,\theta \rangle \vert \mathrm{d}\Xi _{s}(\theta ),\;\;z \in {\mathbb{R}}^{d}$$

(25.3.9)

is the support function of a zonoid with generating measure Ξ _s .

Summarizing all the preceding relations we may formulate the following result.

Theorem 25.3.1.

Each zonoid $\mathcal{Z}$ generates a negative definite kernel on ℝ^d

$$\mathcal{L}_{\mathcal{Z}}(x,y) = h_{\mathcal{Z}}(x - y) = \int\nolimits_{{\mathbb{S}}^{d-1}}^{}\vert \langle x - y,\theta \rangle \vert \mathrm{d}\mu _{\mathcal{Z}}(\theta ).$$

(25.3.10)

This kernel is strongly negative definite if the support of $\mu _{\mathcal{Z}}$ coincides with the whole sphere ${\mathbb{S}}^{d-1}$, and

$$\begin{array}{rcl} \mathcal{N}(\mu, \nu )& =& 2\int\nolimits_{{\mathbb{R}}^{d}}^{} \int\nolimits_{{\mathbb{R}}^{d}}^{}\mathcal{L}_{\mathcal{Z}}(x,y)\mathrm{d}\mu (x)\mathrm{d}\nu (y) -\int\nolimits_{{\mathbb{R}}^{d}}^{} \int\nolimits_{{\mathbb{R}}^{d}}^{}\mathcal{L}_{\mathcal{Z}}(x,y)\mathrm{d}\mu (x)\mathrm{d}\mu (y) \\ & & -\int\nolimits_{{\mathbb{R}}^{d}}^{} \int\nolimits_{{\mathbb{R}}^{d}}^{}\mathcal{L}_{\mathcal{Z}}(x,y)\mathrm{d}\nu (x)\mathrm{d}\nu (y) \\ \end{array}$$

is the square of a distance between measures $\mu, \nu \in \mathcal{B}_{\mathcal{L}}$. This distance has the following representation:

$$\mathfrak{N}(\mu, \nu ) ={ \left (\int\nolimits_{{\mathbb{S}}^{d-1}}^{}\mathrm{d}\mu _{\mathcal{Z}}(\theta )\int\nolimits_{-\infty }^{\infty }{\bigl (F_{ \theta }(a) - G_{\theta }{(a)\bigr )}}^{2}\mathrm{d}a\right )}^{1/2},$$

(25.3.11)

where

$$\begin{array}{rlrlrl} \mu (\mathcal{A}) & = \mathrm{Pr}\{X \in \mathcal{A}\},\;\nu (\mathcal{A}) = \mathrm{Pr}\{Y \in \mathcal{A}\}, & & \\ F_{\theta }(a) & = \mathrm{Pr}\{\langle X,\theta \rangle < a\},\;G_{\theta }(a) = \mathrm{Pr}\{\langle Y,\theta \rangle \, < a\}. &\end{array}$$

(25.3.12)

According to Example 25.3.2, the function $h_{\mathcal{Z}}(u)$ from (25.3.10) may be represented in the form (25.3.3). Let us investigate the connection between $\mu _{\mathcal{Z}}$ in (25.3.10) and Θ in (25.3.3). To do so, we will use the following identity:

$$\vert z\vert = \frac{2} {\pi }\int\nolimits_{0}^{\infty }{\bigl (1 -\cos (zt)\bigr )}\frac{\mathrm{d}t} {{t}^{2}}.$$

(25.3.13)

We have

$$\begin{array}{rcl} h_{\mathcal{Z}}(u)& =& \frac{2} {\pi }\int\nolimits_{{\mathbb{S}}^{d-1}}^{} \int\nolimits_{0}^{\infty }{\bigl (1 -\cos \langle u,\theta \rangle \bigr )}\frac{\mathrm{d}t} {{t}^{2}} \mathrm{d}\mu _{\mathcal{Z}}(\theta ) \\ & =& \frac{2} {\pi }\int\nolimits_{{\mathbb{R}}^{d}}^{}{\bigl (1 -\cos \langle u,v\rangle \bigr )}\frac{1 +\| {v\|}^{2}} {\|{v\|}^{2}} \mathrm{d}\Theta (v).\end{array}$$

So

$$\begin{array}{rlrlrl} \mathrm{d}\Theta (v) & = \frac{2} {\pi } \frac{1} {1 + {t}^{2}}\mathrm{d}t\mathrm{d}\mu (\theta ), & & \\ v & = t \cdot \theta, \;\;\theta \in {\mathbb{S}}^{d-1},\;t \geq 0. &\end{array}$$

(25.3.14)

If $h_{\mathcal{Z}}(u)$ is a support function of a zonoid $\mathcal{Z}$, then clearly

$$h_{\mathcal{Z}}(\tau \cdot u) = \tau h_{\mathcal{Z}}(u)$$

for all τ > 0 and u ∈ ℝ^d, and, as was shown previously, $h_{\mathcal{Z}}(x - y)$ is a negative definite kernel. The inverse is also true.

Theorem 25.3.2.

Suppose that f is a continuous function on ℝ^d such that f(0) = 0, $f(-u) = f(u)$. Then the following facts are equivalent:

• Fact 1. f(τ ⋅ u) = τf(u) and f(x − y) is a negative definite kernel.

• Fact 2. f is a support function of a zonoid.

Proof.

Previously we saw that Fact 2 implies Fact 1, and we must prove only that Fact 1 implies Fact 2. According to Example 25.3.2,

$$f(u) = \int\nolimits_{{\mathbb{R}}^{d}}^{}{\bigl (1 -\cos \langle u,v\rangle \bigr )}\mathrm{d}\Theta _{1}(v),$$

(25.3.15)

where

$$\mathrm{d}\Theta _{1}(v) = \frac{1 +\| {v\|}^{2}} {\|{v\|}^{2}} \mathrm{d}\Theta (v),$$

and Θ is the measure from (25.3.3).

We have

$$f(\tau \cdot u) = \tau f(u)$$

(25.3.16)

for any τ > 0, u ∈ ℝ^d. Substituting (25.3.15) into (25.3.16) and using the uniqueness of the measure Θ in (25.3.3) we obtain

$$\begin{array}{rcl} \int\nolimits_{{\mathbb{R}}^{d}}^{}{\bigl (1 -\cos \langle \tau \cdot u,v\rangle \bigr )}\mathrm{d}\Theta _{1}(v)& =& \tau \int\nolimits_{{\mathbb{R}}^{d}}^{}{\bigl (1 -\cos \langle u,v\rangle \bigr )}\mathrm{d}\Theta _{1}(v), \\ {\bigl (1 -\cos \langle u,v\rangle \bigr )}\mathrm{d}\Theta _{1}(v/\tau )& =& \tau \int\nolimits_{{\mathbb{R}}^{d}}^{}{\bigl (1 -\cos \langle u,v\rangle \bigr )}\mathrm{d}\Theta _{1}(v) \\ \end{array}$$

and

$$\Theta _{1}(v/\tau ) = \tau \Theta _{1}(v).$$

We write here v = r ⋅w for r > 0 and $w \in {\mathbb{S}}^{d-1}$. We have

$$\Theta _{1}(r\tau \cdot w) = \tau \Theta _{1}(r \cdot w)$$

and, finally, for τ = r,

$$\Theta _{1}(r \cdot w) = \frac{1} {r}\Theta _{1}(w).$$

(25.3.17)

It is clear that representation (25.3.15) for Θ₁ of the form (25.3.17) coincides with (25.3.14).²

Note that the $\mathfrak{N}$-distance can be bounded by the Hausdorf distance. Let $\mathcal{Z}_{\mu }$ and $\mathcal{Z}_{\nu }$ be two zonoids with generating measures μ and ν, respectively. The following inequality holds for their supporting functions $h(\mathcal{Z}_{\mu },u)$ and $h(\mathcal{Z}_{\nu },u)$:

$$\vert h(\mathcal{Z}_{\mu },u) - h(\mathcal{Z}_{\nu },u)\vert \leq d_{H}(\mathcal{Z}_{\mu },\mathcal{Z}_{\nu }).$$

Obviously, from this inequality it follows that

$$\mathcal{N}(\mu, \nu ) \leq 2d_{H}(\mathcal{Z}_{\mu },\mathcal{Z}_{\nu }),$$

and therefore

$$\mathfrak{N}(\mu, \nu ) \leq {(2d_{H}(\mathcal{Z}_{\mu },\mathcal{Z}_{\nu }))}^{1/2}.$$

(25.3.18)

Note that each $\mathfrak{N}$-distance generated by a zonoid is an ideal distance of degree 1 ∕ 2.