Gluing hyperconvex metric spaces

TL;DR

This paper explores methods for combining hyperconvex metric spaces through specific gluing techniques, establishing criteria that preserve hyperconvexity and demonstrating limitations of these methods.

Contribution

It introduces two new criteria for gluing hyperconvex spaces along certain subsets and shows these methods are mutually exclusive.

Findings

Gluing along strongly convex subsets preserves hyperconvexity.

Gluing along externally hyperconvex subsets preserves hyperconvexity.

The two gluing methods cannot be combined in a single construction.

Abstract

We investigate how to glue hyperconvex (or injective) metric spaces such that the resulting space remains hyperconvex. We give two new criteria, saying that on the one hand gluing along strongly convex subsets and on the other hand gluing along externally hyperconvex subsets leads to hyperconvex spaces. Furthermore, we show by an example that these two cases where gluing works are opposed and cannot be combined.