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