On diversities and finite dimensional Banach spaces

Abstract

A diversity $\delta$ in $M$ is a function defined over every finite set of points of $M$ mapped onto $[0,\infty )$, with the properties that $\delta (X)=0$ if and only if $|X|\le 1$ and $\delta (X\cup Y)\le \delta (X\cup Z)+\delta (Z\cup Y)$, for every finite sets $X,Y,Z\subset M$ with $|Z|\ge 1$. Its importance relies in the fact that, amongst others, they generalize the notion of metric distance. Our main contribution is the characterization of Banach-embeddable diversities $\delta$ defined over $M$, $|M|=3$, i.e. when there exist points $p_i\in {\mathbb{R}}^n$, $i=1,2,3$, and a symmetric, convex, and compact set $C\subset {\mathbb{R}}^n$ such that $\delta (\{x_{i_1},\dots ,x_{i_m}\})=R(\{p_{i_1},\dots ,p_{i_m}\},C)$, where $R(X,C)$ denotes the circumradius of $X$ with respect to $C$.