Generic Fréchet differentiability of convex functions dominated by a lower semicontinuous convex function (Q1270674)

It is shown that a lower semi-continuous convex function \(f:E\to\mathbb R\cup\{+\infty\}\) with \(f^{-1}(\mathbb R)\neq\emptyset\) has the property that every such function \(g\) with \(g\leq f\) is Fréchet differentiable on a dense \(G_\delta\)-subset of the interior of \(g^{-1}(\mathbb R)\) if and only if the \(w^*\)-closed convex hull of the image of the subdifferential map of \(f\) has the Radon-Nikodým property. The proof uses and extends a (so far) unpublished theorem in [the first named authors, ``Generic Fréchet differentiability of convex functions on non-Asplund spaces''].