Generic differentiability of convex functions on the dual of a Banach space
We study a class of Banach spaces which have the property that every continuous convex function on an open convex subset of the dual possessing a weak\(^*\) continuous subgradient at points of a dense \(G_\delta\) subset of its domain, is Fréchet differentiable on a dense \(G_\delta\) subset of its domain. A smaller more amenable class consists of Banach spaces where every minimal weak\(^*\) cusco from a complete metric space into subsets of the second dual which intersect the embedding from a residual subset of the domain is single-valued and norm upper semicontinuous on a residual subset of the domain. It is known that all Banach spaces with the Radon-Nikodym property belong to these classes as do all with equivalent locally uniformly rotund norm. We show that all with an equivalent weakly locally uniformly rotund norm belong to these classes. The condition closest to a characterization is that the Banach space have its weak topology fragmentable by a metric whose topology on bounded sets is stronger than the weak topology. We show that the space \(\ell_\infty(\Gamma)\), where \(\Gamma\) is uncountable, does not belong to our special classes.
scientific article; zbMATH DE number 910266