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Mazur's intersection property and a Kreĭn-Milman type theorem for almost all closed, convex and bounded subsets of a Banach space


Author: Pando Grigorov Georgiev
Journal: Proc. Amer. Math. Soc. 104 (1988), 157-164
MSC: Primary 46B20
DOI: https://doi.org/10.1090/S0002-9939-1988-0958059-3
MathSciNet review: 958059
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Abstract: Let $ \mathcal{V}$ (resp. $ {\mathcal{V}^*}$) be the set of all closed, convex and bounded (resp. $ {w^*}$-compact and convex) subsets of a Banach space $ E$ (resp. of its dual $ {E^*}$) furnished with the Hausdorff metric. It is shown that if there exists an equivalent norm $ \vert\vert \cdot \vert\vert$ in $ E$ with dual $ \vert\vert \cdot \vert{\vert^*}$ such that $ \left( {E,\vert\vert \cdot \vert\vert} \right)$ has Mazur's intersection property and $ \left( {{E^*},\vert\vert\cdot\vert{\vert^*}} \right)$ has $ {w^*}$-Mazur's intersection property, then

(1) there exists a dense $ {G_\delta }$ subset $ {\mathcal{V}_0}$ of $ \mathcal{V}$ such that for every $ X \in {\mathcal{V}_0}$ the strongly exposing functionals form a dense $ {G_\delta }$ subset of $ {E^*}$;

(2) there exists a dense $ {G_\delta }$ subset $ \mathcal{V}_0^*$ of $ {\mathcal{V}^*}$ such that for every $ {X^*} \in \mathcal{V}_0^*$ the $ {w^*}$-strongly exposing functionals form a dense $ {G_\delta }$ subset of $ E$. In particular every $ X \in {\mathcal{V}_0}$ is the closed convex hull of its strongly exposed points and every $ {X^*} \in \mathcal{V}_0^*$ is the $ {w^*}$-closed convex hull of its $ {w^*}$-strongly exposed points.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1988-0958059-3
Keywords: Strongly exposed points, strongly exposing functionals, closed convex hull, Mazur's intersection property, sublinear functionals, Fréchet differentiability
Article copyright: © Copyright 1988 American Mathematical Society

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