On sets of non-differentiability of Lipschitz and convex functions

Abstract

We observe that each set from the system $\widetilde{\mathcal A}$ (or even $\widetilde{\mathcal{C}}$) is $\Gamma $-null; consequently, the version of Rademacher’s theorem (on Gâteaux differentiability of Lipschitz functions on separable Banach spaces) proved by D. Preiss and the author is stronger than that proved by D. Preiss and J. Lindenstrauss. Further, we show that the set of non-differentiability points of a convex function on ${\mathbb{R}}^n$ is $\sigma $-strongly lower porous. A discussion concerning sets of Fréchet non-differentiability points of continuous convex functions on a separable Hilbert space is also presented.