The Krein-Mil'man Theorem for Metric Spaces with a Convex Bicombing

Abstract

We use bicombings on arcwise connected metric spaces to give definitions of convex sets and extremal points. These notions coincide with the customary ones in the classes of normed vector spaces and geodesic metric spaces which are convex in the usual sense. A rather straightforward modification of the standard proof of the Krein-Mil'man Theorem yields the result that in a large class of metric spaces every compact convex set is the closed convex hull of its extremal points. The result appears to be new even for CAT(0)-spaces.