Ball Intersection Properties in Metric Spaces

TL;DR

This paper explores the properties of hyperconvexity in complete metric spaces, establishing equivalences and conditions that generalize previous results and provide new insights into the structure of hyperconvex subsets.

Contribution

It proves that 4-hyperconvexity is equivalent to finite hyperconvexity in complete spaces and introduces local-to-global results for hyperconvex subsets.

Findings

4-hyperconvexity equals finite hyperconvexity in complete spaces

Complete, almost n-hyperconvex spaces are n-hyperconvex

Conditions for hyperconvex subsets to be convex with geodesic bicombing

Abstract

We show that in complete metric spaces, $4$-hyperconvexity is equivalent to finite hyperconvexity. Moreover, every complete, almost $n$-hyperconvex metric space is $n$-hyperconvex. This generalizes among others results of Lindenstrauss and answers questions of Aronszajn-Panitchpakdi. Furthermore, we prove local-to-global results for externally and weakly externally hyperconvex subsets of hyperconvex metric spaces and find sufficient conditions in order for those classes of subsets to be convex with respect to a geodesic bicombing.