Differentiability of convex functions and the convex point of continuity property in Banach spaces (Q1108539)

It is shown that, for a separable Banach space X, every continuous convex Gateaux differentiable function on X is Fréchet differentiable on a dense set iff \(X^*\) has the weak*-convex point-of-continuity property (i.e. every weak*-compact convex subset of \(X^*\) has a point at which the relative weak* and norm topologies coincide).