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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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