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

1 Equations d'evolution
SAMM - Statistique, Analyse et Modélisation Multidisciplinaire (SAmos-Marin Mersenne)
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 $\Phi$-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.
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Type de document :
Pré-publication, Document de travail
2016
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https://hal-paris1.archives-ouvertes.fr/hal-01308517
Contributeur : Mohammed Bachir <>
Soumis le : dimanche 10 juillet 2016 - 01:18:43
Dernière modification le : lundi 27 novembre 2017 - 14:14:02
Document(s) archivé(s) le : mardi 11 octobre 2016 - 10:24:45

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• HAL Id : hal-01308517, version 4
• ARXIV : 1604.08473

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Mohammed Bachir. On the Krein-Milman-Ky Fan theorem for convex compact metrizable sets.. 2016. 〈hal-01308517v4〉

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