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- W4299440644 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." @default.
- W4299440644 created "2022-10-02" @default.
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- W4299440644 date "2018-01-01" @default.
- W4299440644 modified "2023-09-25" @default.
- W4299440644 title "On the Krein-Milman-Ky Fan theorem for convex compact metrizable sets." @default.
- W4299440644 hasPublicationYear "2018" @default.
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