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Mazur’s intersection property and a Krein-Milman type theorem for almost all closed, convex and bounded subsets of a Banach space. (English) Zbl 0673.46005
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. In this paper, it is shown that if there exists an equivalent norm $$\| \cdot \|$$ in E with dual $$\| \cdot \|^*$$ such that $$(E,\| \cdot \|)$$ has Mazur’s intersection property and $$(E,\| \cdot \|^*)$$ has $$w^*$$-Mazur’s intersection property, then
(i) 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;
(ii) 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.
The author has a question: What are the necessary and sufficient conditions for E $$(E^*)$$ to be an almost Asplund (almost weak Asplund) space?
Reviewer: Yu Xin tai

##### MSC:
 46B20 Geometry and structure of normed linear spaces
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