On Diversities and Finite Dimensional Banach Spaces

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 $|X|\leq 1$ and $\delta(X\cup Y)\leq\delta(X\cup Z)+\delta(Z\cup Y)$, for every finite sets $X,Y,Z\subset M$ with $|Z|\geq 1$. Its importance relies in the fact that, amongst others, they generalize the notion of metric distance.
[1mm] We characterize when a diversity $\delta$ defined over $M$, $|M|=3$, is Banach-embeddable, i.e. when there exist points $p_i$, $i=1,2,3$, and a symmetric, convex, and compact set $C$ 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 $C$. Moreover, we also characterize when a diversity $\delta$ is a Banach diversity, i.e. when $\delta(X)=R(X,C)$, for every finite set $X\subset\mathbb R^n$, where $C$ is an $n$-dimensional, symmetric, convex, and compact set.