Ball intersection properties in metric spaces

Abstract

The goal of the present work is to introduce new metric space techniques to study the intersection properties of families of balls. These new techniques add, on the one hand, to results due to Lindenstrauss on the extension of uniformly continuous functions and compact linear operators, and answer, on the other hand, questions raised by Aronszajn and Panitchpakdi on hyperconvex metric spaces. The present work is divided into two parts. In the first part, the proofs of our two main results on ball intersection properties in metric spaces are presented. The first main result states that for any integer [Formula: see text] greater or equal to three, a complete and almost[Formula: see text]-hyperconvex metric space is automatically [Formula: see text]-hyperconvex. The second main result shows that in complete metric spaces, the property of being [Formula: see text]-hyperconvex is equivalent to the property of being finitely hyperconvex and that the analogues are true for externally[Formula: see text]-hyperconvex and weakly externally[Formula: see text]-hyperconvex subsets. This last result and the results proved later in this work unify the analysis of those three properties: hyperconvexity, external hyperconvexity and weak external hyperconvexity. In the second part of this work, we make the link with notions of convexity and bicombings and present applications. We extend local-to-global results on hyperconvex spaces to externally hyperconvex spaces as well as to weakly externally hyperconvex spaces. We conclude with applications of our results to the characterization of externally hyperconvex and weakly externally hyperconvex subsets of hyperconvex spaces.