More on the differentiability of convex functions

Abstract:

Let $C$ be a closed, convex set in a topological vector space $X$ such that $NS(C)$, the set of its nonsupport points, is nonempty (this is always the case if $X$ is Banach separable; if $X$ is Fréchet, $NS\left ( C \right )$ is residual in $C$). If $X$ is normed, we prove that any locally Lipschitz, convex real function $f$ on $C$ is subdifferentiable on $NS\left ( C \right )$. If in addition $X$ is Banach separable, we prove that $f$ is smooth on a residual subset of $NS\left ( C \right )$.