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 Gδ subset of its domain, is Fréchet differentiable on a dense Gδ 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 semi-continuous at the points of 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 characterisation 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 ℓ∞(Γ), where Γ is uncountable, does not belong to our special
classes.
