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 \({{\,\textrm{CAT}\,}}(0)\) 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 \(A=\{x_1,\ldots ,x_n\}\subset Y\) in a complete \({{\,\textrm{CAT}\,}}(0)\) space Y there exists a Lipschitz map \(f:X\rightarrow Y\) such that the convex hull of \(A\) is contained in f(X).