On the Krein-Milman theorem for convex compact metrizable sets
Résumé
The Krein-Milman theorem states that every convex compact subset of a Hausdorff locally convex topological space, is the closed convex hull of its extreme points. We prove that, in the metrizable case the situation is rather better. Indeed, we introduce a concept of "{\it affine exposed points}" which is intermediate between the notions of exposed points and extreme points. Then, we prove that every convex compact metrizable subset of a Hausdorff locally convex topological space, is the closed convex hull of its affine exposed points. This fails in general for not metrizable compact convex subsets.
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