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arXiv · 2212.10967

On diversities and finite dimensional Banach spaces

Bernardo González Merino
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Abstract

A diversity $δ$ in $M$ is a function defined over every finite set of points of $M$ mapped onto $[0,\infty)$, with the properties that $δ(X)=0$ if and only if $|X|\leq 1$ and $δ(X\cup Y)\leqδ(X\cup Z)+δ(Z\cup Y)$, 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. Our main contribution is the characterization of Banach-embeddable diversities $δ$ 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 $δ(\{x_{i_1},\dots,x_{i_m}\})=R(\{p_{i_1},\dots,p_{i_m}\},C)$, where $R(X,C)$ denotes the circumradius of $X$ with respect to $C$.

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Bernardo González Merino. 2023-02-13. On diversities and finite dimensional Banach spaces. https://arxiv.org/abs/2212.10967

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