Generic Frechet-differentiability of convex functions on small sets
Let C be a convex set in a Banach space E not contained in a closed hyperplane but having no interior points. Suppose C is Baire in the norm topology. Let \(\phi\) : \(C\to {\mathbb{R}}\) be a lower semi-continuous convex function. We ask under which conditions \(\phi\) is generically Fréchet- differentiable on C. In contrast with the classical case int \(C\neq \emptyset\), one first has to settle the problem when \(\partial \phi (x)\neq \emptyset\) generically. This is the case if and only if \(\phi\) is locally Lipschitz at the points of a dense subset of C. Taking this as an assumption and choosing E to be an Asplund space, \(\phi\) turns out to be Fréchet-differentiable at the points of a dense \(G_{\delta}\)-subset of C. In contrast with the case int \(C\neq \emptyset\), Fréchet- differentiability of \(\phi\) at \(x\in C\) does not imply the uniqueness of \(\partial \phi (x)\). The question when, under the above assumptions, uniqueness of \(\partial \phi (x)\) may be achieved generically, naturally leads to the additional requirement that C have at least one non-support point. By a classical result of Klee's, the latter is the case for separable C, but need not be satisfied in general.
Frechet-differentiable at the points of a dense G sub(delta)-subset
scientific article; zbMATH DE number 4090188