A non-compact convex hull in general non-positive curvature

Abstract

In this article, we are interested in metric spaces that satisfy a weak non-positive curvature condition in the sense that they admit a conical bicombing. Recently, these spaces have begun to be studied in more detail, and a rich theory is beginning to emerge. We contribute to this study by constructing a complete metric space X with a conical bicombing σ such that there is a finite subset of X whose closed σ-convex hull is non-compact. In CAT(0)-geometry, the analogous statement is an open question, i.e. it is not known whether closed convex hulls of finite subsets of complete CAT(0) space are compact or not. This question goes back to Gromov. Our result shows that to obtain a positive answer to Gromov's question, more than just the convexity properties of the metric must be used. The constructed space X has the additional property that there is an integer n such that it is an initial object in the category of convex hulls of n-point sets. Thus, roughly speaking, X can be thought of as the largest possible convex hull of n points.