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

Abstract

Gromov’s (open) question whether the closed convex hull of finitely many points in a complete CAT(0) space is compact naturally extends to weaker notions of non-positive curvature in metric spaces. In this article, we consider metric spaces admitting a conical geodesic bicombing, and show that the question has a negative answer in this setting. Specifically, for each n > 1, we construct a complete metric space X admitting a conical geodesic bicombing, which is the closed convex hull of n points and is not compact. The space X moreover has the universal property that for any n points A = {x1,..., xn} ⊂ Y in a complete CAT(0) space Y there exists a Lipschitz map f : X → Y such that the convex hull of A is contained in f (X).