On the structure of Banach spaces with Mazur's intersection property
We establish structural properties for Banach spaces where every bounded closed convex set is the intersection of closed balls and for those with dual space where every bounded \(weak^*\) closed convex set is the intersection of dual balls. It is shown that if a Banach space has both properties then the duality mapping is a homeomorphism between certain residual subsets of the unit spheres of the space and its dual, the points of these sets being simultaneously points of Fréchet differentiability of the norm and strongly exposed points of the unit ball. Among spaces with the second property is the special class where every point of the unit sphere is a denting point of the closed unit ball. For Banach spaces isomorphic to a space of this class, and this includes all the weakly compactly generated spaces, it is shown that every continuous convex function on an open convex subset of the dual is Fréchet differentiable on a dense \(G_{\delta}\) subset of its domain provided the set of points where the function has a weak\({}^*\) continuous subgradient is residual in its domain.
Banach spaces where every bounded closed convex set is the intersection of closed balls
dual space where every bounded \(weak^ *\) closed convex set is the intersection of dual balls
points of Fréchet differentiability of the norm
strongly exposed points of the unit ball
scientific article; zbMATH DE number 4198798