Local differentiability of distance functions

Abstract:

Recently Clarke, Stern and Wolenski characterized, in a Hilbert space, the closed subsets $C$ for which the distance function $d_{C}$ is continuously differentiable everywhere on an open “tube” of uniform thickness around $C$. Here a corresponding local theory is developed for the property of $d_{C}$ being continuously differentiable outside of $C$ on some neighborhood of a point $x\in C$. This is shown to be equivalent to the prox-regularity of $C$ at $x$, which is a condition on normal vectors that is commonly fulfilled in variational analysis and has the advantage of being verifiable by calculation. Additional characterizations are provided in terms of $d_{C}^{2}$ being locally of class $C^{1+}$ or such that $d_{C}^{2}+\sigma |\cdot |^{2}$ is convex around $x$ for some $\sigma >0$. Prox-regularity of $C$ at $x$ corresponds further to the normal cone mapping $N_{C}$ having a hypomonotone truncation around $x$, and leads to a formula for $P_{C}$ by way of $N_{C}$. The local theory also yields new insights on the global level of the Clarke-Stern-Wolenski results, and on a property of sets introduced by Shapiro, as well as on the concept of sets with positive reach considered by Federer in the finite dimensional setting.