On the Krein-Milman theorem for convex compact metrizable sets

Mohammed Bachir (1)

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The Krein-Milman theorem states that a 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, every convex compact metrizable subset of a Hausdorff locally convex topological space, is the closed convex hull of its exposed points. This fails in general for not metrizable compact convex subsets.

Mots clés: Keyword; phrase: Extreme points; exposed points; Krein-Milman theorem; variational principle; 2010 Mathematics Subject:; 46B22; 46B20; 49J50

Domaines: Analyse fonctionnelle [math.FA]

Mohammed Bachir. On the Krein-Milman theorem for convex compact metrizable sets. 2016. ⟨hal-01308517v2⟩