On the Krein-Milman-Ky Fan theorem for convex compact metrizable sets
Abstract
The Krein-Milman theorem (1940) states that every convex compact subset of a Hausdorfflocally convex topological space, is the closed convex hull of its extreme points. In 1963, Ky Fan extended the Krein-Milman theorem to the general framework of -convexity. Under general conditions on the class of functions , the Krein-Milman-Ky Fan theorem asserts then, that every compact -convex subset of a Hausdorff space, is the -convex hull of its -extremal points. We prove in this paper that, in the metrizable case the situation is rather better. Indeed, we can replace the set of -extremal points by the smaller subset of -exposed points. We establish under general conditions on the class of functions , that every -convex compact metrizable subset of a Hausdorff space, is the -convex hull of its -exposed points. As a consequence we obtain that each convex weak compact metrizable (resp. convex weak* compact metrizable) subset of a Banach space (resp. of a dual Banach space), is the closed convex hull of its exposed points (resp. the weak* closed convex hull of its weak* exposed points). This result fails in general for compact -convex subsets that are not metrizable.