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On the Krein-Milman-Ky Fan theorem for convex compact metrizable sets
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
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