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) 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 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 A = x 1, …, x n ⊂ Y in a complete {\, CAT\, } (0) CAT (0) space Y there exists a Lipschitz map f: X Y f: X → Y such that the convex hull of A A is contained in f (X).