Linear and Sublinear Diversities

David Bryant and Paul Tupper

Abstract

Diversities are an extension of the concept of a metric space, where a non-negative value is assigned to every finite set of points, rather than just pairs. A general theory of diversities has been developed which exhibits many deep analogies to metric space theory but also veers off in new directions. Just as many of the most important aspects of metric space theory involve metrics defined on $\mathbb{R}^{k}$ , many applications of diversity theory require a specialized theory for diversities defined on $\mathbb{R}^{k}$ , as we develop here. We focus on two fundamental classes of diversities defined on $\mathbb{R}^{k}$ : those that are Minkowski linear and those that are Minkowski sublinear. Many well-known functions in convex analysis belong to these classes, including diameter, circumradius and mean width. We derive surprising characterizations of these classes, and establish elegant connections between them. Motivated by classical results in metric geometry, and connections with combinatorial optimization, we then examine embeddability of finite diversities into $\mathbb{R}^{k}$ . We prove that a finite diversity can be embedded into a linear diversity exactly when it has negative type and that it can be embedded into a sublinear diversity exactly when it corresponds to a generalized circumradius.

1 Introduction

A diversity [6] is a pair $(X,\delta)$ where $X$ is a set and $\delta$ is a non-negative function defined on finite subsets of $X$ satisfying

(D1) $\delta(A)\geq 0$ and $\delta(A)=0$ if and only if $|A|\leq 1$ ,
(D2) $\delta(A\cup C)\leq\delta(A\cup B)+\delta(B\cup C)$ whenever $B\neq\emptyset$ .

As such, diversities are set-based analogues of metric spaces, and in fact the restriction of a diversity to pairs is a metric space [6].

Properties (D1) and (D2) are equivalent to (D1) together with monotonicity

(D3) $\delta(A)\leq\delta(B)$ whenever $A\subseteq B$

and subadditivity on intersecting sets

(D4) $\delta(A\cup B)\leq\delta(A)+\delta(B)$ when $A\cap B\neq\emptyset$ .

We say $(X,\delta)$ is a semidiversity if (D1) is relaxed to

(D1^′) $\delta(A)\geq 0$ and $\delta(A)=0$ if $|A|\leq 1$ .

That is, sets with two or more points may have zero diversity. This terminology is analogous to at least some of the definitions of semimetrics.

Many well-known set functions are diversities: the diameter of a set; the length of a connecting Steiner tree; the circumradius; the length of a minimal traveling salesperson tour; the mean width; the size of a smallest enclosing zonotope. Two set functions which fail to be diversities are genetic diversity

\pi(A)=\mbox{$\binom{|A|}{2}^{-1}$}\sum_{a,b\in A}d(a,b),

which is not monotonic (D3), and volume of convex hull, which fails (D2) and (D4).

There are broad classes of diversities just like there are broad classes of metrics. The $\ell_{1}$ metrics have the form

d_{1}(a,b)=\sum_{i}|a_{i}-b_{i}|,

while $\ell_{1}$ diversities [7] have the form

\delta_{1}(A)=\sum_{i}\max_{a,b\in A}|a_{i}-b_{i}|.

Negative-type metrics satisfy

\sum_{a,b}x_{a}x_{b}\,d(a,b)\leq 0

for all zero-sum vectors $x$ while negative type diversities [27] satisfy

\sum_{A,B}x_{A}x_{B}\,\delta(A\cup B)\leq 0

for all zero-sum vectors $x$ with $x_{\emptyset}=0$ .

The theory of diversities sometimes runs in parallel with that of metric spaces and other times veers off in new directions. In the first paper on diversities [6] we explored how concepts of hyperconvexity, injectivity and the tight span extended, and to an extent enriched, the analogous metric concepts. In [7] we showed that the ‘geometry of graphs’ [18] linking metric embeddings to approximation algorithms on graphs has a parallel ‘geometry of hypergraphs’ linking diversity embeddings to approximation algorithms on hypergraphs. Jozefiak and Shephard [17] use this approach to obtain the best known approximation algorithms for several hypergraph optimization problems.

Diversities turn out to be an exemplary class of metric structures, exhibiting fascinating connections with model theory and Urysohn’s universal space [4, 5, 15, 14]. Other directions that have been pursued are a diversity analogue of ultrametric and normed spaces [13, 20, 12] and new diversity-based results in fixed point theory [9, 22].

Our focus here is on the intersection of diversity theory, geometry and convex analysis. Recall that the Minkowski sum of two subsets $A,B\subseteq\mathbb{R}^{k}$ is given by $A+B=\{a+b:a\in A,b\in B\}.$ We investigate diversities defined on $\mathbb{R}^{k}$ which are (Minkowski) linear [24]

(D5) $\delta(\lambda A)=\lambda\delta(A)\mbox{ and }\delta(A+B)=\delta(A)+\delta(B)$

and those which are (Minkowski) sublinear

(D6) $\delta(\lambda A)=\lambda\delta(A)\mbox{ and }\delta(A+B)\leq\delta(A)+\delta(B),$

for $\lambda\geq 0$ and $A,B$ nonempty finite subsets of $\mathbb{R}^{k}$ . Many familiar diversities defined on $\mathbb{R}^{k}$ are Minkowski linear or sublinear (see below). We explore their properties and characterization.

As per usual, we make repeated use of support functions when dealing with convex bodies and functions defined on them. The support function of a nonempty bounded set $A$ is defined

h_{A}:\mathbb{R}^{k}\rightarrow\mathbb{R}:x\mapsto\sup\{a\cdot x:a\in A\}.

Here $a\cdot x$ denotes the usual dot product in $\mathbb{R}^{k}$ ,

a\cdot x=\sum_{i=1}^{k}a_{i}x_{i}.

We note that a set has the same support function as both its closure and convex hull.

We make use of the following properties of the support function, see [24, Chapter 1] for further details.

1.

$h_{A+B}=h_{A}+h_{B}$ and $h_{\lambda A}=\lambda h_{A}$ for for non-empty, bounded $A,B$ and $\lambda\geq 0.$
2.

If $A,B$ are nonempty convex compact sets then $A\subseteq B$ if and only $h_{A}(x)\leq h_{B}(x)$ for all $x\in\mathbb{R}^{k}$
3.

A function $h:\mathbb{R}^{k}\rightarrow\mathbb{R}$ is the support function for some bounded nonempty set if and only if $h(x+y)\leq h(x)+h(y)$ and $h(\lambda x)=\lambda h(x)$ for all $x,y\in\mathbb{R}^{k}$ and $\lambda\geq 0$ (that is, $h$ is sublinear).

We often consider support functions restricted to $\mathbb{S}^{k-1}$ , the unit sphere in $\mathbb{R}^{k}$ , noting that a support function is determined everywhere by its values on $\mathbb{S}^{k-1}$ . We note that the support function restricted to $\mathbb{S}^{k-1}$ of a nonempty set is bounded if and only if the set is bounded.

Our main results for diversities and semidiversities $(\mathbb{R}^{k},\delta)$ are:

1.

(Theorem 5) Linear diversities and semidiversities are exactly those which can be written in the form

$\delta(A)=\int_{\mathbb{S}^{k-1}}h_{A}(x)\mathrm{d}\nu(x)$

for a Borel measure $\nu$ on the sphere $\mathbb{S}^{k-1}$ satisfying

$\int_{\mathbb{S}^{k-1}}x\,\mathrm{d}\nu(x)=0.$ (1)
2.

(Theorem 7) The extremal linear semidiversities are those where the support of $\nu$ is a finite, affinely independent set, which in turn correspond to a generalized circumradius (a Minkowski semidiversity) based on the simplex.
3.

(Theorem 8) A diversity or semidiversity is sublinear if and only if it is the maximum of linear semidiversities (just like a function is convex if and only if it is the maximum of linear functions).

We then shift to studying the embeddings of finite diversities into linear or sublinear diversities. Questions regarding embeddings and approximate embeddings of metrics in normed spaces are central to metric geometry and its applications. Consider, for example, Menger’s characterizations of when a metric can be embedded in Euclidean space, or the vast literature applying metric embeddings to combinatorial optimizations (reviewed in [19, 8] and [16]).

For finite diversities $(X,\delta)$ we show:

1.

(Theorem 11) A finite diversity can be embedded in a linear diversity if and only if it has negative type, meaning that

$\sum_{A,B}x_{A}x_{B}\delta(A\cup B)\leq 0$

for all vectors $x$ with zero sum and $x_{\emptyset}=0$ .
2.

(Theorem 12) A finite diversity can be embedded as a sublinear diversity if and only if it can be embedded in a Minkowski diversity (that is, a generalized circumradius) if and only if it is the maximum of a collection of negative type diversities.

2 Linear and sublinear diversities

In this section we establish basic properties and characterizations for linear and sublinear diversities.

2.1 Examples of Linear and Sublinear Diversities

We start with examples of diversities which are linear or sublinear. Note that for all diversities $(X,\delta)$ we have $\delta(\emptyset)=0$ , even if that is not stated explicitly below.

1.

Let $\|\cdot\|$ be any norm on $\mathbb{R}^{k}$ . The diameter diversity is given by

$\delta(A)=\max_{a,b\in A}\|a-b\|$

for finite $A\subseteq\mathbb{R}^{k}$ . The diameter diversity is sublinear [24, pg 49].

The $\ell_{1}$ diversity $(\mathbb{R}^{k},\delta_{1})$ is

\delta_{1}(A)=\sum_{i=1}^{k}\max_{a,b\in A}(a_{i}-b_{i}).

for finite $A\subseteq\mathbb{R}^{k}$ [7]. For finite $A,B$ and $\lambda\geq 0$ we have

	$\displaystyle\delta_{1}(\lambda A+B)$	$\displaystyle=\sum_{i=1}^{k}\max\left\{\left((\lambda a+b)_{i}-(\lambda a^{\prime}+b^{\prime})_{i}\right):a,a^{\prime}\in A,\,b,b^{\prime}\in B\right\}$
		$\displaystyle=\sum_{i=1}^{k}\max\{\lambda(a_{i}-a_{i}^{\prime})+(b_{i}-b_{i}^{\prime}):a,a^{\prime}\in A,\,b,b^{\prime}\in B\}$
		$\displaystyle=\lambda\delta_{1}(A)+\delta_{1}(B).$

so $(\mathbb{R}^{k},\delta_{1})$ is a linear diversity.

3.

The circumradius of finite $A\subset\mathbb{R}^{k}$ with respect to the unit ball $\mathcal{B}$ is

$\delta(A)=\min\{\lambda\geq 0:A\subseteq\lambda\mathcal{B}+x\mbox{ for some $x\in\mathbb{R}^{k}$}\}.$

More generally, the Minkowski diversity $(\mathbb{R}^{k},\delta_{K})$ with kernel $K$ is equal to the generalized circumradius

$\delta_{K}(A)=\inf\{\lambda\geq 0:A\subseteq\lambda K+x\mbox{ for some }x\in\mathbb{R}^{k}\},$

for finite $A\subseteq\mathbb{R}^{k}$ . Minkowski diversities are sublinear [3] but are not, in general, linear. For example, consider the circumradius diversity $(\mathbb{R}^{2},\delta)$ . If $A=\{(0,0),(1,0)\}$ and $B=\{(0,0),(0,1)\}$ then $\delta(A+B)<\delta(A)+\delta(B)$ .

We assume that $K$ is closed, convex and has non-empty interior. We have elsewhere required that the kernel $K$ be bounded, however in this paper we will not require this. Note that if $K$ is an unbounded, $(\mathbb{R}^{k},\delta_{K})$ is a semidiversity rather than a diversity.
4.

The mean-width diversity $(\mathbb{R}^{k},\delta_{w})$ is

$\delta_{w}(A)=\frac{2}{\omega_{k}}\int_{\mathbb{S}^{k-1}}h_{A}(x)\,d\nu(x)$

where $\nu(x)$ is the (uniform) Haar measure on the sphere and $\omega_{k}=\int_{\mathbb{S}^{k-1}}\,d\nu(x)$ . Equivalently, $\delta_{w}(A)$ is the mean-width of the convex hull of $A$ . Mean-width diversities are linear [24, pg 50].

Let $w_{A}(x)=\max\{x\cdot(a-b):a,b\in A\}$ denote the width of $A$ in direction $x$ , so $w_{A}(x)=h_{A-A}(x)=h_{A}(x)+h_{A}(-x)$ . Then

$\delta_{w}(A)=\frac{1}{\omega_{k}}\int_{\mathbb{S}^{k-1}}w_{A}(x)\,d\nu(x).$

For $1\leq p<\infty$ we define

$\delta^{(p)}_{w}(A)=\frac{1}{\omega_{k}}\left[\int_{\mathbb{S}^{k-1}}|w_{A}(x)|^{p}\,d\nu(x)\right]^{1/p}.$

That this is a sublinear diversity follows from the Minkowski inequality. See [13] Proposition 2.4, or [2] Proposition 10 in the case that $p=2$ .
5.

A zonotope $Z$ is a Minkowski sum of line segments and the length $\ell(Z)$ of the zonotope equals the sum of the length of the line segments. We define the zonotope diversity $(\mathbb{R}^{k},\delta_{z})$ where $\delta_{z}(A)$ is the minimum length of a zonotope containing $A$ . We show that zonotope diversities are sublinear in Proposition 2.

In a Euclidean space $\mathbb{R}^{k}$ , any non-negative linear combination of sublinear semidiversities is sublinear, and any non-negative linear combination of linear semidiversities is linear. Hence the set of sublinear semidiversities forms a convex cone, as does the set of linear semidiversities.

2.2 Properties of linear and sublinear diversities

We establish some basic properties of sublinear diversities (and hence of linear diversities). This includes the continuous extension of sublinear diversities from finite sets to bounded sets.

Proposition 1.

Let $\delta$ be a function on finite subsets of $\mathbb{R}^{k}$ which satisfies (D1), monotonicity (D3) and sublinearity (D6).

1.

$\delta$ is translation invariant: $\delta(A+x)=\delta(A)$ for all finite $A\subseteq\mathbb{R}^{k}$ and $x\in\mathbb{R}$ .
2.

$(\mathbb{R}^{k},\delta)$ is a diversity.
3.

If $\mathrm{conv}(A)=\mathrm{conv}(B)$ then $\delta(A)=\delta(B)$ .
4.

The map $N:\mathbb{R}^{k}\rightarrow\mathbb{R}$ given by $N(x)=\delta(\{0,x\})$ is a norm on $\mathbb{R}^{k}$ .
5.

For all finite $A\subseteq\mathbb{R}^{k}$ with $|A|>2$ we have

$\delta(A)\leq\mbox{ $\frac{|A|-1}{|A|(|A|-2)}$}\sum_{a\in A}\delta(A\setminus\{a\}).$

If $\delta$ satisfies (D1^′) rather than (D1) then 1-5 still hold except that $(\mathbb{R}^{k},\delta)$ is a semidiversity and $N$ is a seminorm.

Proof.

By sublinearity (D6) and (D1), we have

	$\displaystyle\delta(A+x)$	$\displaystyle\leq\delta(A)+\delta(\{x\})=\delta(A),\mbox{ and }$
	$\displaystyle\delta(A)$	$\displaystyle\leq\delta(A+x)+\delta(\{-x\})=\delta(A+x).$

As $\delta$ is monotonic and $\delta(\emptyset)=0$ , $\delta$ is non-negative, and by part 1. $\delta$ is translation invariant. We show that $(X,\delta)$ satisfies (D4). Suppose that $x\in A\cap B$ . Then $0\in(A-x)\cap(B-x)$ and so $(A-x)\cup(B-x)\subseteq(A-x)+(B-x)$ and

	$\displaystyle\delta(A\cup B)$	$\displaystyle=\delta\Big{(}(A-x)\cup(B-x)\Big{)}$
		$\displaystyle\leq\delta\Big{(}(A-x)+(B-x)\Big{)}$
		$\displaystyle\leq\delta(A-x)+\delta(B-x)$
		$\displaystyle=\delta(A)+\delta(B).$

Hence $(\mathbb{R}^{k},\delta)$ satisfies (D1), (D3) and (D4).

3.

Proposition 2.2b in [3].
4.

By (D5) we have $N(x+y)=\delta(\{0,x+y\})\leq\delta(\{0,x\})+\delta(\{0,y\})=N(x)+N(y)$ . If $\lambda>0$ then $N(\lambda x)=\delta(\{0,\lambda x\})=\lambda\delta(0,x)=|\lambda|N(x)$ , while if $\lambda<0$ we have

$N(\lambda x)=\delta(\{\lambda x,0\})=\delta(\{0,-\lambda x\})=|\lambda|N(x).$

Also, $N(0)=\delta(\{0\})=0$ if and only if $\delta(\{x,0\})=0$ if and only if $x=0$ .
5.

This follows from sublinearity and the following observation; see the proof of [1, Theorem 4.1]. By sublinearity we may assume $\sum_{a\in A}a=0$ . So for each $a\in A$

$-\frac{1}{|A|-1}a=\frac{1}{|A|-1}\sum_{a^{\prime}\neq a}a^{\prime}\in\mathrm{conv}(A\setminus\{a\}).$

We also have $a\in\mathrm{conv}(A\setminus a^{\prime})$ for $a\neq a^{\prime}$ . So for all $a\in A$

$\frac{(|A|-2)|A|}{|A|-1}a=(|A|-1)a-\frac{1}{|A|-1}a\in\sum_{b\in A}\mathrm{conv}(A\setminus\{b\}).$

This gives

$A\subseteq\frac{|A|-1}{|A|(|A|-2)}\sum_{a\in A}\mathrm{conv}(A\setminus\{a\})$

and applying sublinearity gives the result.

∎

It is now straightforward to show that the zonotope diversity introduced above is in fact a sublinear diversity.

Proposition 2.

The zonotope diversity $(\mathbb{R}^{k},\delta_{z})$ is a sublinear diversity.

Proof.

Recall that $\delta_{z}(A)$ is the shortest length of a zonotope containing $A$ . The function $\delta_{z}(A)$ is clearly monotonic, vanishes when $|A|\leq 1$ , and is strictly positive when $|A|>1$ . Given finite $A,B$ , let $Z_{A}$ and $Z_{B}$ the the minimum length zonotopes containing $A$ and $B$ respectively. Then $Z_{A}+Z_{B}$ is a zonotope containing $A+B$ with length $\ell(Z_{A})+\ell(Z_{B})$ . By Proposition 1 part 2, $(\mathbb{R}^{k},\delta_{z})$ is a sublinear diversity. ∎

The zonotope diversity is not linear: let $A=\{(0,0),(1,0),(0,1)\}$ and $B=-A$ . Then $\delta_{z}(A)=\delta_{z}(B)=2$ but $\delta_{z}(A+B)=2+\sqrt{2}$ .

In a semidiversity, (D1) is replaced by (D1^′), and sets with more than one element can have diversity zero. When the semidiversity is sublinear, the sets with zero diversity are highly structured. Define the null set of a semidiversity $(\mathbb{R}^{k},\delta)$ to be the set

\mathrm{null}(\delta)=\Big{\{}x:\delta(\{0,x\})=0\Big{\}}

and $\mathrm{null}(\delta)^{\perp}=\{x\in\mathbb{R}^{k}:x\cdot y=0\mbox{ for all }y\in\mathrm{null}(\delta)\}$ .

Proposition 3.

Let $(\mathbb{R}^{k},\delta)$ be a sublinear semidiversity.

1.

$\mathrm{null}(\delta)$ is a linear subspace of $\mathbb{R}^{k}$
2.

$\delta$ restricted to $\mathrm{null}(\delta)^{\perp}$ is a diversity
3.

If $P$ is the projection operator for $\mathrm{null}(\delta)^{\perp}$ then $\delta(A)=\delta(PA)$ for all finite $A\subseteq\mathbb{R}^{k}$ .

Proof.

1.

For $x,y\in\mathrm{null}(\delta)$ and $\alpha>0$ we have $\delta(\{0,x+y\})\leq\delta(\{0,x\})+\delta(\{0,y\})=0$ and $\delta(\{0,\alpha x\})=\alpha\delta(\{0,x\})=0$ so $x+y\in\mathrm{null}(\delta)$ and $\alpha x\in\mathrm{null}(\delta)$ . By translation invariance, $\delta(\{0,-x\})=\delta(\{x,0\})=0$ and $-x\in\mathrm{null}(\delta)$ .
2.

Suppose $x,y\in\mathrm{null}(\delta)^{\perp}$ and $\delta(\{x,y\})=0$ . We have that $x-y\in\mathrm{null}(\delta)^{\perp}$ by part 1. By translation invariance $\delta(\{0,x-y\})=\delta(\{x,y\})=0$ which implies $x-y\in\mathrm{null}(\delta)$ . Hence $x-y$ is both in a subspace and its orthogonal complement, and so $x=y$ .
3.

For all finite $A\subseteq\mathbb{R}^{k}$ we have $A\subseteq PA+B$ and $PA\subseteq A+C$ for some $B,C\subset\mathrm{null}(\delta)$ . We have $\delta(B)=0$ since

$0\leq\delta(B)\leq\sum_{b\in B}\delta(\{0,b\})=0,$

and, likewise, $\delta(C)=0$ . By sublinearity, $\delta(A)=\delta(PA)$ .

∎

Let $\|\cdot\|$ be a norm on $\mathbb{R}^{k}$ with associated metric $d(x,y)=\|x-y\|$ and unit ball $\mathcal{B}=\{x:\|x\|\leq 1\}$ . The Hausdorff distance between two nonempty closed bounded sets $K$ and $L$ can be defined by [24, p. 61] :

d_{H}(K,L)=\min\{\lambda:K\subseteq L+\lambda\mathcal{B}\mbox{ and }L\subseteq K+\lambda\mathcal{B}\}.

For bounded $K\subseteq\mathbb{R}^{k}$ define

\delta^{*}(K)=\sup\{\delta(A):A\subseteq K\mbox{ finite}\}.

(2)

Proposition 4.

Let $(\mathbb{R}^{k},\delta)$ be a sublinear semidiversity.

1.

For all bounded $K\subseteq\mathbb{R}^{k}$ , $\delta^{*}(K)<\infty$ .
2.

For all finite $A\subseteq\mathbb{R}^{k}$ we have

$\delta^{*}(\mathrm{conv}(A))=\delta(A).$
3.

For all bounded $K,L\subset\mathbb{R}^{k}$ and $\lambda\geq 0$

$\delta^{*}(K+L)\leq\delta^{*}(K)+\delta^{*}(L)$

and

$\delta^{*}(\lambda K)=\lambda\delta^{*}(K).$
4.

If $(\mathbb{R}^{k},\delta)$ is linear then the restriction of $\delta^{*}$ to the set of nonempty compact convex subsets of $\mathbb{R}^{k}$ is a valuation. That is,

$\delta^{*}(K\cap L)+\delta^{*}(K\cup L)=\delta^{*}(K)+\delta^{*}(L)$

for all nonempty compact convex bodies $K,L$ such that $K\cap L$ and $K\cup L$ are non-empty and convex.
5.

The restriction of $\delta^{*}$ to the set of nonempty compact convex subsets of $\mathbb{R}^{k}$ is Lipschitz continuous with respect to the Hausdorff metric, with Lipschitz constant $\delta^{*}(\mathcal{B})$ .

Proof.

1.

By equivalency of norms on $\mathbb{R}^{k}$ we have that $K$ is bounded with respect to metric $d$ if and only if it is bounded with respect to the induced metric of $\delta$ . Let $V$ be the set of vertices of some polytope (e.g. a cube) containing $K$ . For all finite $A\subseteq K$ we have by monotonicity and part 2 that

$\delta(A)\leq\delta(A\cup V)=\delta(V)$

so that $\delta^{*}(K)\leq\delta(V)<\infty.$
2.

Let $A$ be a finite subset of $\mathbb{R}^{k}$ and let $K=\mathrm{conv}(A)$ . For any $A^{\prime}\subseteq K$ we have $\mathrm{conv}(A\cup A^{\prime})=\mathrm{conv}(A)$ so $\delta(A^{\prime})\leq\delta(A\cup A^{\prime})=\delta(A)$ by Proposition 1 (ii). Hence

$\delta(A)\leq\sup\{\delta(A^{\prime}):\mbox{ finite }A^{\prime}\subseteq K)\}\leq\delta(A).$
3.

Fix $\epsilon>0$ and suppose that $C$ is a finite subset of $K+L$ such that $\delta(C)>\delta^{*}(K+L)-\epsilon$ . For each $c\in C$ there is $a_{c}\in K$ and $b_{c}\in L$ such that $c=a_{c}+b_{c}$ . Let $A=\{a_{c}:c\in C\}\subseteq K$ and $B=\{b_{c}:c\in C\}\subseteq L$ so that $C\subseteq A+B$ . It follows that

$\delta^{*}(K+L)-\epsilon<\delta(C)\leq\delta(A+B)\leq\delta(A)+\delta(B)\leq\delta^{*}(K)+\delta^{*}(L).$

Taking $\epsilon$ to zero gives the result.

Let $A$ be a finite subset of $K$ such that $\delta(A)>\delta^{*}(K)-\epsilon$ . As $\lambda A\subseteq\lambda K$ we have

$\lambda(\delta^{*}(K)-\epsilon)<\lambda\delta(A)=\delta(\lambda A)\leq\delta^{*}(\lambda K).$

Hence $\delta^{*}(\lambda K)\geq\lambda\delta^{*}(K)$ from which equality follows by symmetry.
4.

By Lemma 3.1.1 of [24] we have that if $K,L,K\cup L$ and $K\cap L$ are nonempty compact convex subsets then

$(K\cup L)+(K\cap L)=K+L.$

By linearity,

$\delta^{*}(K\cup L)+\delta^{*}(K\cap L)=\delta^{*}(K)+\delta^{*}(L).$

Suppose that $K,L$ are bounded nonempty subsets satisfying $d_{H}(K,L)=\lambda$ . For any $\epsilon>0$ there is a finite $A\subseteq K$ such that $\delta(A)\leq\delta^{*}(K)<\delta(A)+\epsilon$ . We also have $A\subseteq K\subseteq L+\lambda\mathcal{B}$ so there is finite $B\subseteq L$ and $C\subseteq\mathcal{B}$ such that $A\subseteq B+\lambda C$ . Hence

\delta^{*}(K)<\delta(A)+\epsilon\leq\delta(B)+\lambda\delta(C)+\epsilon\leq\delta^{*}(L)+\lambda\delta^{*}(\mathcal{B})+\epsilon.

By a symmetric argument,

\delta^{*}(L)<\delta^{*}(K)+\lambda\delta^{*}(\mathcal{B})+\epsilon.

Taking $\epsilon$ to zero, we have

|\delta^{*}(K)-\delta^{*}(L)|\leq\delta^{*}(\mathcal{B})d_{H}(K,L).

The bound is tight, as can be seen by letting $K=\mathcal{B}$ and $L=2\mathcal{B}$ . Then $d_{H}(K,L)=1$ and $|\delta^{*}(K)-\delta^{*}(L)|=\delta^{*}(\mathcal{B})$ .

∎

Bryant et al. [3] also describe an extension of Minkowski diversities from finite sets to bounded sets. They define $\widetilde{\delta}(P)=\delta(\mathrm{vert}(P))$ for any polytope with vertex set $\mathrm{vert}(P)$ , and extend that to general bounded convex sets $K$ by defining $\widetilde{\delta}(K)=\lim_{n\rightarrow\infty}\widetilde{\delta}(P_{n})$ for any sequence of polytopes $P_{1},P_{2},\ldots$ converging to $K$ . Proposition 4 part 2. gives that $\delta^{*}(P)=\widetilde{\delta}(P)$ for any polytope, while from Proposition 4 part 5 we have that $\delta^{*}(K)=\widetilde{\delta}(K)$ . Hence $\delta^{*}$ coincides with $\widetilde{\delta}$ for Minkowski diversities.

2.3 Characterization of linear diversities

The following characterization of linear diversities is essentially contained in the proof of the main theorem in Firey [10]; see also [21].

Theorem 5.

Let $\delta$ be a function defined on finite subsets of $\mathbb{R}^{k}$ . Then $(X,\delta)$ is a linear semidiversity if and only if there is a positive finite Borel measure $\nu$ on the unit sphere $\mathbb{S}^{k-1}=\{x\in\mathbb{R}^{k}:\|x\|_{2}=1\}$ such that

\int_{\mathbb{S}^{k-1}}x\,\mathrm{d}\nu(x)=0

(3)

and

\delta(A)=\int_{\mathbb{S}^{k-1}}h_{A}(x)\,\mathrm{d}\nu(x)

(4)

for all finite $A\subseteq\mathbb{R}^{k}$ . Such a measure is unique.

Proof.

First we show that $\delta$ given by (4) is a linear semidiversity. For $a\in\mathbb{R}^{k}$ and finite $A\subseteq B\subseteq\mathbb{R}^{k}$ we have

\displaystyle\delta(\{a\})

\displaystyle=\int_{\mathbb{S}^{k-1}}h_{\{a\}}(x)\,\mathrm{d}\nu(x)=\int_{\mathbb{S}^{k-1}}a\cdot x\,\mathrm{d}\nu(x)=a\cdot\int_{\mathbb{S}^{k-1}}x\,\mathrm{d}\nu(x)=0

and, since $h_{A}(x)\leq h_{B}(x)$ and $h_{A+B}(x)=h_{A}(x)+h_{B}(x)$ for all $x$ we have $\delta(A)\leq\delta(B)$ and $\delta(A+B)=\delta(A)+\delta(B)$ . By Proposition 1, $(\mathbb{R}^{k},\delta)$ is a linear semidiversity.

For the converse, let $(\mathbb{R}^{k},\delta)$ be a linear semidiversity and define $\delta^{*}$ as in (2). By Proposition 4 the restriction of $\delta^{*}$ to nonempty compact convex subsets is Minkowski linear, monotonic and vanishes on singletons. From the proof of the main theorem in [10], we have that, for all compact convex sets $K$ ,

\delta^{*}(K)=\int_{\mathbb{S}^{k-1}}h_{K}(x)\,\mathrm{d}\nu(x)

for some positive finite Borel measure $\nu$ satisfying (3), and $\nu$ is the unique such measure. (See [23, Thm 2.14] for details on the use of the Riesz Theorem in this case.) Now for any nonempty finite $A$ , let $K=\mathrm{conv}(A)$ . Since $\delta(A)=\delta^{*}(K)$ and $h_{A}=h_{K}$ , the result follows for all nonempty finite $A$ . ∎

Refer to caption — Figure 1: Support of measures corresponding to (a) mean width; (b) the $L_{1}$ diversity; and (c) a Minkowski diversity with a simplex kernel.

In Figure 1 we depict the support for the measures corresponding to mean width (uniform on the unit circle), the $L_{1}$ diversity ( $\pm e_{i}$ ), and the Minkowski diversity for a simplex kernel. The first two of these are easy enough to demonstrate. We prove the third example below, after we have a characterization of extremal linear diversities.

2.4 Extremal linear diversities

The set of linear semidiversities on $\mathbb{R}^{k}$ forms a cone. A non-zero semidiversity $\delta$ is extremal (or lies on an extremal ray) if it cannot be expressed as the convex combination of two linear semidiversities which are not its scale copies. We make use of Theorem 5 to characterize the extremal linear diversities and semidiversities. First we prove a technical result simplifying evaluation of the Minkowski diversity for a simplex.

Lemma 6.

Let $v_{0},\ldots,v_{j}\in\mathbb{R}^{k}$ be affinely independent with $\sum_{\ell=0}^{j}c_{\ell}v_{\ell}=0$ , for some $c_{\ell}\geq 0$ , $\sum_{\ell}c_{\ell}=1$ . Define the polyhedron $K=\{y:v_{\ell}\cdot y\leq 1,\,\text{for all }\ell\}$ . Let $\delta_{K}$ be the Minkowski semidiversity given by $K$ . Then

\delta_{K}(A)=\sum_{\ell=0}^{j}c_{\ell}h_{A}(v_{\ell}).

for all finite $A\subseteq\mathbb{R}^{k}$ .

Proof.

Let $A=\{a_{i}\}_{i=1,\ldots,|A|}$ . We express $\delta_{K}(A)$ as the solution to a linear program. Recall that $\delta_{K}(A)$ is the minimum $\lambda$ such that there is some $x\in\mathbb{R}^{k}$ such that $a_{i}-x\in\lambda K$ for all $i$ . We can rewrite this constraint as $v_{\ell}\cdot(a_{i}-x)\leq\lambda$ for all $i,\ell$ . If we take $\lambda$ and $x$ to be our primal variables we get the following linear program:

	minimize	$\displaystyle\lambda=(1,0)\cdot(\lambda,x)$
	subject to	$\displaystyle\lambda+v_{\ell}\cdot x\geq v_{\ell}\cdot a_{i},\mbox{for all }i\mbox{ and }\ell.$

The dual linear program with dual variables $y_{i\ell}$ is

	maximize	$\displaystyle\sum_{i\ell}(v_{\ell}\cdot a_{i})y_{ij}$
	subject to	$\displaystyle y_{i\ell}\geq 0,\mbox{for all }i\mbox{ and }\ell,$
		$\displaystyle\sum_{i\ell}y_{i\ell}=1,$
		$\displaystyle\sum_{i\ell}v_{\ell}y_{i\ell}=0.$

Let $\bar{y}_{\ell}=\sum_{i}y_{i\ell}$ . Then our dual constraints are equivalent to

\sum_{\ell}\bar{y}_{\ell}=1,\ \ \ \sum_{\ell}\bar{y}_{\ell}v_{\ell}=0.

Since the $v_{\ell}$ are affinely independent and $\sum_{\ell}c_{\ell}v_{\ell}=0$ , $\sum_{\ell}c_{\ell}=1$ , there is a unique solution given by $\bar{y}_{\ell}=c_{\ell}$ for all $\ell$ . Now it remains to determine for each $\ell$ the value of $y_{i\ell}$ for each $i$ . We need to maximize $\sum_{i\ell}(v_{\ell}\cdot a_{i})y_{i\ell}$ given $y_{i\ell}\geq 0$ and $\sum_{i}y_{i\ell}=c_{\ell}$ . For each $\ell$ , the solution is to let $y_{i\ell}=c_{\ell}$ for the $i$ that maximizes $(v_{\ell}\cdot a_{i})$ , and $0$ otherwise. This gives for the solution to the dual problem

\sum_{\ell}c_{\ell}\max_{a\in A}v_{\ell}\cdot a=\sum_{\ell}c_{\ell}h_{A}(v_{\ell}).

∎

The following theorem identifies extremal linear semidiversities as Minkowski diversities $\delta_{K}$ with $K$ equal to a simplex or a simplex plus a subspace.

Theorem 7.

The following are equivalent for a semidiversity $(\mathbb{R}^{k},\delta)$ :

(i)

$(\mathbb{R}^{k},\delta)$ is extremal in the class of linear semidiversities.
(ii)

$(\mathbb{R}^{k},\delta)$ satisfies

$\delta(A)=\int_{\mathbb{S}^{k-1}}h_{A}(x)\,\mathrm{d}\nu(x)$

for all finite $A\subseteq\mathbb{R}^{k}$ , where $\nu$ is a measure on $\mathbb{S}^{k-1}$ with $\int_{\mathbb{S}^{k-1}}x\,\mathrm{d}\nu(x)=0$ , such that the support of $\nu$ is a finite, affinely independent set.
(iii)

$(\mathbb{R}^{k},\delta)$ is a Minkowski semidiversity with kernel $K$ of the form

$K=\mathrm{conv}(W)+H^{\perp},$

where $W$ is an affinely independent set of points, $H$ is the affine closure of $W$ ,and $H^{\perp}$ is the orthogonal space to $H$ .

Proof.

(i) $\Rightarrow$ (ii). Suppose $\delta$ is extremal and the support of $\nu$ is not affinely independent. Let $H$ be the affine hull of the support of $\nu$ , with $\dim H=j$ , and let $S=H\cap\mathbb{S}^{k-1}$ . Affine dependence implies $\nu$ is not supported on only $j+1$ points or fewer. Therefore we can partition $S$ into $S_{1},\ldots,S_{j+2}$ , each with $\nu(S_{i})>0$ . Let $m_{i}=\int_{S_{i}}\,\mathrm{d}\nu(x)=\nu(S_{i})>0$ . Then

m=\int_{S}\,\mathrm{d}\nu(x)=\sum_{i}\int_{S_{i}}d\nu(x)=\sum_{i}m_{i}

and

0=\int_{S}x\,\mathrm{d}\nu(x)=\sum_{j}\int_{S_{i}}x\,\mathrm{d}\nu(x)=\sum_{i}m_{i}x_{i}

where $x_{i}=(\int_{S_{i}}x\,\mathrm{d}\nu(x))/m_{i}$ . Choose a subset of the $x_{i}$ with $k+1$ points so that 0 is in the convex hull of them. Let’s say they are $x_{1},\ldots,x_{k+1}$ . Find $\mu_{i}$ for $i=1,...,k+1$ such that $\sum_{i}\mu_{i}x_{i}=0$ , and $\mu_{i}<m_{i}$ . Let $\mu_{k+2}=0$ . Now define $\nu^{\prime}$ by $\nu^{\prime}(A)=(\mu_{j}/m_{j})\nu(A)$ for $A\subseteq S_{i},j=1...k+1$ , and zero otherwise. Then $\nu^{\prime}\leq\nu$ , and $\nu^{\prime}$ has smaller support, because $m_{k+2}>0$ but $\mu_{k+2}=0$ . Also

\int x\,\mathrm{d}\nu^{\prime}(x)=\sum_{i}\int_{S_{i}}x\frac{\mu_{j}}{m_{j}}d\nu(x)=\sum_{i}\frac{\mu_{i}}{m_{j}}m_{i}x_{i}=\sum_{i}\mu_{i}x_{i}=0.

We can now write $\nu=(\nu-\nu^{\prime})+\nu^{\prime}$ where $\nu-\nu^{\prime}$ and $\nu$ are not scale copies, so $\delta$ is not extremal in the cone of linear semidiversities.
(ii) $\Rightarrow$ (i). Suppose that

\delta(A)=\int_{\mathbb{S}^{k-1}}h_{A}(x)\,\mathrm{d}\nu(x)

for all finite $A\subseteq\mathbb{R}^{k}$ and some measure $\nu$ on $\mathbb{S}^{k-1}$ with affinely independent support. Let $\delta_{1}$ and $\delta_{2}$ be linear semidiversities with corresponding measures $\nu_{1}$ and $\nu_{2}$ . If $\mathrm{supp}(\nu_{1})\subseteq\mathrm{supp}(\nu)$ then affine independence and the constraint that $\int_{\mathbb{S}^{k-1}}x\,\mathrm{d}\nu_{1}(x)=0$ implies that $\nu_{1}$ is a scaled version of $\nu$ . Likewise for $\nu_{2}$ . Hence if $\nu=\lambda\nu_{1}+(1-\lambda)\nu_{2}$ for $\lambda\in[0,1]$ we have that $\mathrm{supp}(\nu_{1})\cup\mathrm{supp}(\nu_{2})\subseteq\mathrm{supp}(\nu)$ and both $\nu_{1}$ and $\nu_{2}$ are scale versions of $\nu$ . This shows that $\delta$ is an extremal linear diversity.
(ii) $\Rightarrow$ (iii). Let the support of $\nu$ be the points $u_{0},\ldots,u_{j}$ with weights $m_{\ell}>0$ such that $\sum_{\ell}m_{\ell}u_{\ell}=0$ . Let $m=\sum_{\ell}m_{\ell}$ , $c_{\ell}=m_{\ell}/m$ , and $v_{\ell}=mu_{\ell}$ , so that $\delta(A)=\sum_{\ell}c_{\ell}h_{A}(v_{\ell})$ , $\sum_{\ell}c_{\ell}v_{\ell}=0$ , and $\sum_{\ell}c_{\ell}=1$ . Let $V=\{v_{0},\ldots,v_{j}\}$ , which is affinely independent because the $u_{\ell}$ are. Let $H$ be the span of $V$ and $H^{\perp}$ its orthogonal complement. By Lemma 6, $\delta=\delta_{K}$ , the Minkowski semidiversity for the set $K=\{y:y\cdot v_{\ell}\leq 1,\mbox{ for all }\ell\}$ . The intersection of $K$ with $H$ is a simplex; let it have vertices $W=\{w_{0},\ldots,w_{j}\}$ . Then $K=\mathrm{conv}(V)+H^{\perp}$ as required.
(iii) $\Rightarrow$ (ii). By translating $K$ if necessary, we may assume $0$ is in the relative interior of $\mathrm{conv}(W)$ . We can write $K=\{y:v_{\ell}\cdot y\leq 1\}$ for some affinely independent $V=\{v_{0},\ldots,v_{j}\}$ . Because $\mathrm{conv}(W)$ is bounded, $0\in\mathrm{conv}(V)$ . So there are $c_{\ell}\geq 0$ with $\sum_{\ell}c_{\ell}v_{\ell}=0$ and $\sum_{\ell}c_{\ell}=1$ . By Lemma 6 we have

\delta_{K}(A)=\sum_{\ell}c_{\ell}h_{A}(v_{\ell})

for all finite $A$ . Let $m_{\ell}=c_{\ell}|v_{\ell}|$ and $u_{\ell}=v_{\ell}/|v_{\ell}|$ . Then the $u_{\ell}$ are also affinely independent. Let $\nu$ be the measure that assigns mass $m_{\ell}$ to each $u_{\ell}$ . ∎

Points in a finite dimensional convex cone can always be written as convex combinations of extremal points. The cone of linear semidiversities has infinite dimensional, so proving that linear semidiversities are in the convex hull of extremal diversities requires a little more work.

Theorem 8.

A semidiversity $(\mathbb{R}^{k},\delta)$ is linear if and only if $\delta$ is a convex combination of extremal linear semidiversity functions.

Proof.

Since a weighted average of linear semidiversities is a linear semidiversity, one way is immediate. For the other, suppose that $\delta$ is a linear semidiversity. By Theorem 5, there is a Borel measure $\nu$ on $\mathbb{S}^{k-1}$ such that $\int x\,\mathrm{d}\nu(x)=0$ and $\delta(A)=\int h_{A}(x)\,\mathrm{d}\nu(x)=0$ for all finite $A$ . Let $m=\int\,\mathrm{d}\nu(x)$ .

Let $E$ be the set of all signed Borel measures on $\mathbb{S}^{k-1}$ , which is a Hausdorff locally convex set [26, p. 134]. The space $C$ of measures on $\mathbb{S}^{k-1}$ with $\int x\,\mathrm{d}\nu(x)=m$ and $\int x\,\mathrm{d}\nu(x)=0$ is compact by the Banach-Alaoglu theorem [26, p. 114], and is convex.

We claim that the set of extremal points of $C$ is closed. Let $\nu_{n}$ , $n\geq 1$ be a sequence of extremal measures that converges in the vague topology, so that $\int f\,\mathrm{d}\nu_{n}$ converges to $\int f\,\mathrm{d}\nu$ for some $\nu\in C$ for all continuous bounded $f$ . By repeatedly taking subsequences, we can obtain a subsequence $\nu_{n_{k}}=\sum_{i=1}^{j}\mu_{i,k}\delta_{x_{i,k}}$ (where $\delta_{x}$ is a unit mass measure at $x$ ) where $x_{i,k}\rightarrow x_{i}$ and $\mu_{i,k}\rightarrow\mu_{i}$ for some $x_{i}\in\mathbb{S}^{k-1}$ and $\mu_{i}\geq 0$ . Since $\int f\,\mathrm{d}\nu_{n_{k}}\rightarrow\int f\,\mathrm{d}\nu$ as $k\rightarrow\infty$ , we must have $\nu=\sum_{i}\mu_{i}\delta_{x_{i}}$ , showing that the limit is also an extremal point in $C$ . Hence the set of extremal measures is closed.

We can apply a version of the Krein-Milman Theorem ([26, Corollary 17.7]) to obtain that $\delta$ is a weighted average of members of the closure of the extreme points of $C$ . Since the set of extremal points of $C$ is closed, the result follows. ∎

2.5 Characterization of sublinear diversities

We now turn our attention to sublinear diversities. We will show that the relationship between sublinear and linear diversities parallels that between convex and linear functions. Just as every convex function is the supremum of linear functions, every sublinear diversity is the supremum of linear diversities (Theorem 9). In fact, in our case, the supremum is attained for each set, so the value of every sublinear diversity on a set is the maximum of the value of a family of linear diversities on the set. Our proof relies heavily on the ‘Sandwich Theorem’ (Theorem 1.2.5) of [11].

Theorem 9.

Let $\delta$ be a function on finite subsets of $\mathbb{R}^{k}$ . If $(\mathbb{R}^{k},\delta)$ is a sublinear diversity or semidiversity then there is a collection $\{(\mathbb{R}^{k},\delta_{\gamma})\}_{\gamma\in\Gamma}$ of linear semidiversities such that

\delta(A)=\max\{\delta_{\gamma}(A):\gamma\in\Gamma\}.

Conversely, for any collection $\{(\mathbb{R}^{k},\delta_{\gamma})\}_{\gamma\in\Gamma}$ of linear semidiversities and $\delta$ defined by $\delta(A)=\sup_{\gamma\in\Gamma}\delta_{\gamma}(A)$ , $(\mathbb{R}^{k},\delta)$ is a sublinear semidiversity.

Proof.

Suppose that $\{(\mathbb{R}^{k},\delta_{\gamma})\}_{\gamma\in\Gamma}$ are linear semidiversities and

\delta(A)=\sup\{\delta_{\gamma}(A):\gamma\in\Gamma\}

for all finite $A\subseteq\mathbb{R}^{k}$ . Note that $\delta$ vanishes on singletons and is monotonic since each $\delta_{\gamma}$ has these properties. Suppose that $A,B$ are finite subsets of $\mathbb{R}^{k}$ and $\lambda\geq 0$ . Then

\delta(\lambda A)=\sup\{\delta_{\gamma}(\lambda A):\gamma\in\Gamma\}=\sup\{\lambda\delta_{\gamma}(A):\gamma\in\Gamma\}=\lambda\delta(A)

and

\delta(A+B)=\sup\{\delta_{\gamma}(A+B):\gamma\in\Gamma\}=\sup\{\delta_{\gamma}(A)+\delta_{\gamma}(B):\gamma\in\Gamma\}\leq\delta(A)+\delta(B).

So $\delta$ is sublinear. By Proposition 1, $\delta$ is a sublinear semidiversity.

For the converse, suppose that $(\mathbb{R}^{k},\delta)$ is sublinear. Define $\mathcal{H}$ to be the set of all support functions $h_{A}$ for nonempty finite $A\subseteq\mathbb{R}^{k}$ . Define $p$ on the convex cone $\mathcal{H}$ by $p(h_{A})=\delta(A)$ for all finite sets $A$ . The function $p$ is sublinear (and convex in the terminology of [25]), as for any finite $A,B$ ,

p(h_{A}+h_{B})=p(h_{A+B})=\delta(A+B)\leq\delta(A)+\delta(B)=p(h_{A})+p(h_{B}),

and $p(\lambda h_{A})=\lambda p(h_{A})$ for $\lambda\geq 0$ .

Fix finite $B\subseteq\mathbb{R}^{k}$ . Define $q_{B}$ on $\mathcal{H}$ by

q_{B}(h_{A})=\sup\{\lambda:\lambda B+x\subseteq\mathrm{conv}(A)\mbox{ for some $x\in\mathbb{R}^{k}$}\}.

That is, $q_{B}(h_{A})$ is the largest we can scale $B$ so that a translate is contained in $\mathrm{conv}(A)$ . Note that $q_{B}(h_{B})=1$ . This tells us that $p(h_{B})=\delta(B)=\delta(B)q_{B}(h_{B})$ .

We show that $q_{B}$ is superlinear. For all $\alpha\geq 0$ we have $q_{B}(\alpha h_{A})=q_{B}(h_{\alpha A})=\alpha q_{B}(h_{A})$ . Now suppose that $A_{1},A_{2}$ are finite and non-empty subsets of $\mathbb{R}^{k}$ . Given $\epsilon>0$ there are $\lambda_{1}>q_{B}(h_{A_{1}})-\epsilon/2$ , $\lambda_{2}>q_{B}(h_{A_{2}})-\epsilon/2$ , $x_{1},x_{2}\in\mathbb{R}^{k}$ such that

	$\displaystyle\lambda_{1}B+x_{1}$	$\displaystyle\subseteq\mathrm{conv}(A_{1})$
	$\displaystyle\lambda_{2}B+x_{2}$	$\displaystyle\subseteq\mathrm{conv}(A_{2})$
and hence
	$\displaystyle(\lambda_{1}+\lambda_{2})B+(x_{1}+x_{2})$	$\displaystyle\subseteq\mathrm{conv}(A_{1})+\mathrm{conv}(A_{2})$
		$\displaystyle=\mathrm{conv}(A_{1}+A_{2}),$

so that $q_{B}(h_{A_{1}+A_{2}})\geq(\lambda_{1}+\lambda_{2})>q_{B}(h_{A_{1}})+q_{B}(h_{A_{2}})-\epsilon$ . Taking $\epsilon\rightarrow 0$ gives superlinearity.

We now have that $p$ is monotonic and sublinear and that $q_{B}$ is superlinear.

Furthermore, for any finite $A\subseteq\mathbb{R}^{k}$ and $\epsilon>0$ there is $\lambda$ such that $q_{B}(h_{A})-\epsilon<\lambda\leq q_{B}(h_{A})$ and $x\in\mathbb{R}^{k}$ such that $\lambda B+x\subseteq\mathrm{conv}(A)$ , and so

	$\displaystyle p(h_{A})$	$\displaystyle\geq p(h_{\lambda B})$
		$\displaystyle=\lambda p(h_{B})$
		$\displaystyle>(q_{B}(h_{A})-\epsilon)\delta(B).$

Taking $\epsilon\rightarrow 0$ we conclude that $q_{B}(h_{A})\delta(B)\leq p(h_{A})$ for all $h_{A}\in\mathcal{H}$ . Recall that $q_{B}(h_{B})\delta(B)=p(h_{B})$ .

For each finite $B$ we have now satisfied the conditions for Theorem 1.2.5 of [11]:

Let $F$ be a pre-ordered cone and let $p:F\rightarrow\overline{\mathbb{R}}$ be monotone and sublinear, $q:F\rightarrow\overline{\mathbb{R}}$ superlinear with $q\leq p$ . Then there is a monotone linear $\mu:F\rightarrow\overline{\mathbb{R}}$ with $q\leq\mu\leq p$ .

In our example $F$ is the cone $\mathcal{H}$ of support functions of finite sets. Let $q(h)=\delta(B)q_{B}(h)$ . Let $\mu_{B}\colon\mathcal{H}\rightarrow\mathbb{R}$ be the linear map given by the theorem. It is monotone, linear, and

\delta(B)q_{B}(h)\leq\mu_{B}(h)\leq p(h).

Since by definition $p(h_{\{a\}})=\delta(\{a\})=0$ for all $a\in\mathbb{R}^{k}$ , $\mu_{B}(h_{\{a\}})=0$ for all $a\in\mathbb{R}^{k}$ .

Now define $\delta_{B}$ by $\delta_{B}(A)=\mu_{B}(h_{A})$ for all finite $A$ . Then $\delta_{B}$ vanishes on singletons, it is monotone, linear, and hence also sublinear. By Proposition 1 $(\mathbb{R}^{k},\delta_{B})$ is a linear semidiversity.

Because $\delta(B)q_{B}(h_{B})=p(h_{B})$ , we have that

\delta_{B}(B)=\mu_{B}(h_{B})=\delta(B)q_{B}(B)=\delta(B)

and for general finite $A$ we have

\delta_{B}(A)=\mu_{B}(h_{A})\leq p(h_{A})=\delta(A).

Repeating this process for all finite $B\subseteq\mathbb{R}^{k}$ we obtain a set of linear semidiversities $\{\delta_{B}\}_{\text{finite }B\subseteq\mathbb{R}^{k}}$ such that $\delta_{B}\leq\delta$ and $\delta_{B}(B)=\delta(B)$ for all finite $B\subseteq\mathbb{R}^{k}$ . So for all finite $A\subseteq\mathbb{R}^{k}$ ,

\delta(A)=\sup\{\delta_{B}(A):\mbox{ finite }B\subseteq\mathbb{R}^{k}\}=\max\{\delta_{B}(A):\mbox{ finite }B\subseteq\mathbb{R}^{k}\},

since the supremum is actually attained when $B=A$ . ∎

3 Embedding into linear and sublinear diversities

We now turn our attention from linear and sublinear diversities to the questions of when finite diversities can be isometrically embedded within linear or sublinear diversities. Questions about embedding of metric spaces have, of course, been central to metric geometry and its applications, particularly after Linial et al. [18] demonstrated the link between approximate embeddings and combinatorial optimization algorithms on graphs. We showed in [7] that an analogous link holds between approximate embeddings of diversities and combinatorial optimization algorithms on hypergraphs. Here we only consider embeddings without distortion, that is, exact rather than approximate embeddings.

A map $f:X_{1}\mapsto X_{2}$ between two diversities $(X_{1},\delta_{1})$ and $(X_{2},\delta_{2})$ is an isometric embedding if $\delta_{2}(f(A))=\delta_{1}(A)$ for all finite $A\subseteq X_{1}$ . We say that a finite diversity $(X,\delta)$ is linear-embeddable if there is an isometric embedding from $(X,\delta)$ to a linear diversity on $\mathbb{R}^{k}$ for some $k$ and sublinear-embeddable if there is an isometric embedding to some sublinear diversity on $\mathbb{R}^{k}$ , for some $k$ . At this stage we allow the dimension $k$ to be arbitrary.

Theorem 11 gives a characterization of linear-embeddability while Theorem 12 gives a characterization of sublinear-embeddability. Minkowski diversities and negative type diversities were reviewed earlier. We first establish a lemma on finite diversities that are embeddable in extremal linear diversities.

Lemma 10.

If $(\mathbb{R}^{k},\delta)$ is an extremal linear semidiversity, and $X\subseteq\mathbb{R}_{k}$ is finite then $(X,\delta)$ is Minkowski embeddable with a simplex kernel.

Proof.

By Theorem 7 $(\mathbb{R}^{k},\delta)$ is the Minkowski semidiversity with kernel $K=\mathrm{conv}(W)+H^{\perp}$ , where $W$ is a set of affinely independent vectors lying in a subspace $H$ . Let $T$ be an orthogonal matrix so that $TH=\mathrm{span}(\{e_{1},\ldots,e_{j}\})$ and $TH^{\perp}=\mathrm{span}(\{e_{j+1},\ldots,e_{j}\})$ . Let $T_{H}$ be the first $j$ rows of $T$ , so that $T_{H}H=\mathbb{R}^{j}$ , $T_{H}H^{\perp}=\{0\}$ and $T_{H}K=\mathrm{conv}(T_{H}W)$ is a full-dimensional simplex in $\mathbb{R}^{j}$ . Then for all $\lambda$ , $A+x\subset\lambda K$ for some $x\in\mathbb{R}^{k}$ if and only if $T_{H}A+y\subset\lambda\mathrm{conv}(T_{H}W)$ for some $y\in\mathbb{R}^{j}$ . So $\delta(A)=\delta_{\mathrm{conv}(T_{H}W)}(T_{H}A)$ for all finite $A$ , as required. ∎

Theorem 11.

Let $(X,\delta)$ be a finite diversity. The following are equivalent:

(i)

$(X,\delta)$ is linear-embeddable.
(ii)

$(X,\delta)$ has negative type.
(iii)

$(X,\delta)$ can be embedded into a Minkowski diversity $(\mathbb{R}^{k},\delta_{K})$ with kernel equal to a simplex $K\subseteq\mathbb{R}^{k}$ .

Proof.

(i) $\Rightarrow$ (ii) Without loss of generality assume $X\subseteq\mathbb{R}^{k}$ where $(\mathbb{R}^{k},\delta)$ is a linear diversity. From Theorem 8 we have that any linear diversity can be expressed as a convex combination of extremal linear semidiversities. By Lemma 10 each of these extremal linear semidiversities can be expressed as a Minkowski diversity with a simplex, each of which has negative type by Theorem 17 in [3]. As the set of negative type diversities forms a convex cone, $(X,\delta)$ also has negative type.
(ii) $\Leftrightarrow$ (iii) This is Theorem 17 in [3].
(iii) $\Rightarrow$ (i) Theorem 8 shows that any Minkowski diversity with a simplex kernel (being a trivial example of a weighted average of such diversities) is linear. Therefore, if $(X,\delta)$ is embeddable in a Minkowski diversity with a simplex kernel, it also embeddable in a linear diversity. ∎

Theorem 12.

Let $(X,\delta)$ be a finite diversity. The following are equivalent:

(i)

$(X,\delta)$ is sublinear-embeddable.
(ii)

$(X,\delta)$ can be embedded into a Minkowski diversity.
(iii)

$(X,\delta)$ is the maximum of a collection of negative type diversities.

Proof.

(ii) $\Rightarrow$ (i) If $(X,\delta)$ is embeddable as a Minkowski diversity, then it is sublinear-embeddable, since by Theorem 2.4 of [3] all Minkowski diversities are sublinear.
(i) $\Rightarrow$ (ii) Let $(X,\delta)$ be a sublinear-embeddable. We may assume $X$ is a subset of $\mathbb{R}^{k}$ where $(\mathbb{R}^{k},\delta)$ is a sublinear diversity. By Theorem 9 there is a family of linear semidiversities $\delta_{\gamma}$ for $\gamma\in\Gamma$ such that $\delta(A)=\max\delta_{\gamma}(A)$ . Since $X$ is finite, it has a finite number of subsets, and so we may assume that $\Gamma$ is finite. By Proposition 4.1 (a) in [3] if two finite diversities are Minkowski embeddable, then so is their maximum, and hence the same is true of any finite number of finite Minkowski embeddable diversities. Therefore $(X,\delta)$ is Minkowski embeddable.
(i) $\Rightarrow$ (iii) We may assume $X\subseteq\mathbb{R}^{k}$ where $(\mathbb{R}^{k},\delta)$ is a sublinear diversity. By Theorem 9 there is a family of linear semidiversities $\delta_{\gamma}$ for $\gamma\in\Gamma$ such that $\delta(A)=\max\delta_{\gamma}(A)$ . Since $X$ is finite, it has a finite number of subsets, and so we may assume that $\Gamma$ is finite. By Theorem 11, each of $(X,\delta_{\gamma})$ is negative type, and therefore $(X,\delta)$ is the maximum of a collection of negative type diversities.
(iii) $\Rightarrow$ (ii) Suppose $(X,\delta)$ is the maximum of a collection of negative type diversities. By Theorem 11 $(X,\delta)$ can then be represented as the maximum of a collection of Minkowski diversities, and since $X$ is finite, we may assume the collection is finite. By Proposition 4.1 (a) in [3] the maximum of a finite collection of Minkowski embeddable diversities is Minkowski embeddable. ∎

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