Diversities, hyperconvexity and fixed points

TL;DR

This paper explores the properties of diversities and hyperconvexity, establishing fixed point results for nonexpansive mappings in bounded diversities and analyzing the relationship with induced metric spaces.

Contribution

It provides new insights into the fixed point properties of hyperconvex diversities and clarifies the connection between diversities and their induced metric spaces.

Findings

Bounded hyperconvex diversities have fixed point property for nonexpansive mappings.

The induced metric space of a hyperconvex diversity may not be hyperconvex.

Boundedness in diversities does not necessarily transfer to the induced metric space.

Abstract

Diversities have been recently introduced as a generalization of metrics for which a rich tight span theory could be stated. In this work we take up a number of questions about hyperconvexity, diversities and fixed points of nonexpansive mappings. Most of these questions are motivated by the study of the connection between a hyperconvex diversity and its induced metric space for which we provide some answers. Examples are given, for instance, showing that such a metric space need not be hyperconvex but still we prove, as our main result, that they enjoy the fixed point property for nonexpansive mappings provided the diversity is bounded and that this boundedness condition cannot be transferred from the diversity to the induced metric space.