A non-compact convex hull in generalized 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 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.