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 c ontinuous convex function on an open convex subset of the dual possess ing a weak continuous subgradient at points of a dense G(delta) subs et of its domain, is Frechet differentiable on a dense G(delta) subset of its domain. A smaller more amenable class consists of Banach space s where every minimal weak cusco from a complete metric space into s ubsets of the second dual which intersect the embedding from a residua l subset of the domain is single-valued and norm upper semi-continuous at the points of a residual subset of the domain. It is known that al l Banach spaces with the Radon-Nikodym property belong to these classe s as do all with equivalent locally uniformly rotund norm. We show tha t all with an equivalent weakly locally uniformly rotund norm belong t o these classes. The condition closest to a characterisation is that t he Banach space have its weak topology fragmentable by a metric whose topology on bounded sets is stronger than the weak topology. We show t hat the space l(infinity)(Gamma), where Gamma is uncountable, does not belong to our special classes.