Subsmooth sets: Functional characterizations and related concepts

Abstract

Prox-regularity of a set (Poliquin-Rockafellar-Thibault, 2000), or its global version, proximal smoothness (Clarke-Stern-Wolenski, 1995) plays an important role in variational analysis, not only because it is associated with some fundamental properties as the local continuous differentiability of the function dist ( C ; ⋅ ) \mbox {dist}\,(C;\cdot ) , or the local uniqueness of the projection mapping, but also because in the case where C C is the epigraph of a locally Lipschitz function, it is equivalent to the weak convexity (lower-C 2 ^{2} property) of the function. In this paper we provide an adapted geometrical concept, called subsmoothness , which permits an epigraphic characterization of the approximate convex functions (or lower-C 1 ^{1} property). Subsmooth sets turn out to be naturally situated between the classes of prox-regular and of nearly radial sets. This latter class has been recently introduced by Lewis in 2002. We hereby relate it to the Mifflin semismooth functions.