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

Abstract

AbstractGromov’s (open) question whether the closed convex hull of finitely many points in a complete $${{\,\textrm{CAT}\,}}(0)$$ 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 $$A=\{x_1,\ldots ,x_n\}\subset Y$$ A = { x 1 , … , x n } ⊂ Y in a complete $${{\,\textrm{CAT}\,}}(0)$$ CAT ( 0 ) space Y there exists a Lipschitz map $$f:X\rightarrow Y$$ f : X → Y such that the convex hull of $$A$$ A is contained in f(X).