Characterizations of Prox-Regular Sets in Uniformly Convex Banach Spaces

The aim of the paper is to extend to the setting of uniformly convex Banach spaces the results obtained for prox-regular sets in Hilbert spaces. Prox-regularity of a set C at a point x of C is a variational condition related to normal vectors and which is common to many types of sets. In the context of uniformly convex Banach spaces, the prox-regularity of a closed set C at x is shown to be still equivalent to the property of the distance function dC to be continuously differentiable outside of C on some neighbourhood of x. Additional characterizations are provided in terms of metric projection mapping. We also examine the global level of prox-regularity corresponding to the continuous differentiability of the distance function dC over an open tube of uniform thickness around the set C.