Local differentiability of distance functions

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 dC being continuously differentiable outside of C on some neighborhood of a point x is an element of 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 C1+ or such that d(C)(2) + sigma\.\(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.