Local differentiability of distance functions

Recently Clarke, Stern and Wolenski characterized, in a Hilbert space, the closed subsets C C for which the distance function d C dC 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 dC being continuously differentiable outside of C C on some neighborhood of a point x ∈ C x∈ 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 dC² being locally of class C 1 + C¹⁺ or such that d C 2 + σ | ⋅ | 2 dC²+σ |· |² is convex around x x for some σ > 0 σ >0