Local differentiability of distance functions

Source abstract

Recently Clarke, Stern and Wolenski characterized, in a Hilbert space, the closed subsets C C for which the distance function d C d_{C} is continuously differentiable everywhere on an open “tube” of uniform thickness around C C . Here a corresponding local theory is developed for the property of d C d_{C} being continuously differentiable outside of C C on some neighborhood of a point x ∈ C x\in C . This is shown to be equivalent to the prox-regularity of C C at x 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 d_{C}^{2} being locally of class C 1 + C^{1+} or such that d C 2 + σ | ⋅ | 2 d_{C}^{2}+\sigma |\cdot |^{2} is convex around x x for some σ > 0 \sigma >0 . Prox-regularity of C C at x x corresponds further to the normal cone mapping N C N_{C} having a hypomonotone truncation around x x , and leads to a formula for P C P_{C} by way of N C 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.