Ball Intersection Properties in Metric Spaces

Understand

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.