Abstract

A diversity δ\delta in MM is a function defined over every finite set of points of MM mapped onto [0,∞)[0,\infty), with the properties that δ(X)=0\delta(X)=0 if and only if ∣X∣≤1|X|\leq 1 and δ(X∪Y)≤δ(X∪Z)+δ(Z∪Y)\delta(X\cup Y)\leq\delta(X\cup Z)+\delta(Z\cup Y), for every finite sets X,Y,Z⊂MX,Y,Z\subset M with ∣Z∣≥1|Z|\geq 1. Its importance relies in the fact that, amongst others, they generalize the notion of metric distance.
We characterize when a diversity δ\delta defined over MM, ∣M∣=3|M|=3, is Banach-embeddable, i.e. when there exist points pip_i, i=1,2,3i=1,2,3, and a symmetric, convex, and compact set CC such that δ({xi1,…,xim})=R({pi1,…,pim},C)\delta(\{x_{i_1},\dots,x_{i_m}\})=R(\{p_{i_1},\dots,p_{i_m}\},C), where R(X,C)R(X,C) denotes the circumradius of XX with respect to CC. Moreover, we also characterize when a diversity δ\delta is a Banach diversity, i.e. when δ(X)=R(X,C)\delta(X)=R(X,C), for every finite set X⊂RnX\subset\mathbb R^n, where CC is an nn-dimensional, symmetric, convex, and compact set.

Author information

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Bernardo González Merino

Área de Matemática Aplicada, Dep. de Ingeniería y Tecnología de Computadores, Facultad de Informática, Universidad de Murcia, Murcia, Spain

bgmerino@um.es

Keywords

  • Diversity
  • generalized circumradius
  • Minkowski diversity
  • Minkowski embeddable
  • Banach embeddable

Mathematics Subject Classification

52A20; 52A21, 52A40

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Suggested citation

B. González Merino. “On Diversities and Finite Dimensional Banach Spaces.” Journal of Convex Analysis 32 (2025), No. 4, 1227–1240. https://doi.org/10.68381/jca32064

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Published by Heldermann Verlag, 2025. Rights now held by Banach Press LLC.