More on the differentiability of convex functions
Maria Elena Verona · Proceedings of the American Mathematical Society · 1988
Let C C be a closed, convex set in a topological vector space X X such that N S ( C ) NS(C) , the set of its nonsupport points, is nonempty (this is always the case if X X is Banach separable; if X X is Fréchet, N S ( C ) NS\left ( C \right ) is residual in C C ). If X X is normed, we prove that any locally Lipschitz, convex real function f f on C C is subdifferentiable on N S ( C ) NS\left ( C \right ) . If in addition X X is Banach separable, we prove that f f is smooth on a residual subset of N S ( C ) NS\left ( C \right ) .