Compact sets and the closure of their convex hulls in CAT(0) spaces

Abstract: We study the closure of the convex hull of a compact set in a complete CAT(0) space. First we give characterization results in terms of compact sets and the closure of their convex hulls for locally compact CAT(0) spaces that are either regular or satisfy the geodesic extension property. Later inspired by a geometric interpretation of Carath\'eodory's Theorem we introduce the operation of threading for a given set. We show that threading exhibits certain monotonicity properties with respect to intersection and union of sets. Moreover threading preserves compactness. Next from the commutativity of threading with any isometry mapping we prove that in a flat complete CAT(0) space the closure of the convex hull of a compact set is compact. We apply our theory to the computability of the Fr\'echet mean of a finite set of points and show that it is constructible in at most a finite number of steps, whenever the underlying space is of finite type.