Weakly Externally Hyperconvex Subsets and Hyperconvex Gluings

TL;DR

This paper characterizes when the gluing of hyperconvex metric spaces along certain subsets remains hyperconvex, providing a full description for gluings of identical hyperconvex spaces and exploring finite-dimensional cases.

Contribution

It offers a necessary and sufficient condition for hyperconvex gluings along weakly externally hyperconvex subsets and characterizes convex polyhedra in $l_in^n$ that are weakly externally hyperconvex.

Findings

Gluings of hyperconvex spaces are hyperconvex under specific conditions.

Full characterization of gluings of two isometric hyperconvex spaces.

Identification of weakly externally hyperconvex convex polyhedra in $l_in^n$.

Abstract

We give a necessary and sufficient condition for gluings of hyperconvex metric spaces along weakly externally hyperconvex subsets in order that the resulting space be hyperconvex. This leads to a full characterization of gluings of two isometric copies of the same hyperconvex space. Furthermore, we investigate the case of gluings of finite dimensional hyperconvex linear spaces along linear subspaces. For this purpose, we characterize the convex polyhedra in $l_\infty^n$ which are weakly externally hyperconvex.