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 Hausdorff locally 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 Φ-convexity. Under general conditions on the class of functions Φ, the Krein-Milman-Ky Fan theorem asserts then, that every compact Φ-convex subset of a Hausdorff space, is the Φ-convex hull of its Φextremal points. We prove in this paper that, in the metrizable case the situation is rather better. Indeed, we can replace the set of Φ-extremal points by the smaller subset of Φ-exposed points. We establish under general conditions on the class of functions Φ, that every Φ-convex compact metrizable subset of a Hausdorff space, is the Φ-convex hull of its Φ-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 Φ-convex subsets that are not metrizable.