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

TL;DR

This paper explores metric spaces with weak non-positive curvature, demonstrating that convex hulls of finite sets can be non-compact, which contrasts with classical expectations in non-positive curvature geometry.

Contribution

It introduces a counterexample in weak non-positive curvature spaces showing convex hulls of finite sets need not be compact, challenging previous assumptions.

Findings

Existence of a complete metric space with a conical bicombing

Finite sets with non-compact convex hulls in this space

Counterexample to Gromov's compactness question in this setting

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 geodesic bicombing. We show that the analog of a question of Gromov about compactness properties of convex hulls has a negative answer in this setting. Specifically, we prove that there exists a complete metric space $X$ that admits a conical bicombing $\sigma$ such that $X$ has a finite subset whose closed $\sigma$-convex hull is not compact.