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Subsmooth sets: functional characterizations and related concepts. (English) Zbl 1094.49016
Summary: Prox-regularity of a set [R. A. Poliquin, R. T. Rockafellar and L. Thibault, Trans. Am. Math. Soc. 352, No. 11, 5231–5249 (2000; Zbl 0960.49018)], or its global version, proximal smoothness [F. H. Clarke, R. J. Stern and P. R. Wolenski, J. Convex Anal. 2, No. 1–2, 117–144 (1995; Zbl 0881.49008)] 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 $\text{dist}\phantom{\rule{0.166667em}{0ex}}\left(C;·\right)$, or the local uniqueness of the projection mapping, but also because in the case where $C$ is the epigraph of a locally Lipschitz function, it is equivalent to the weak convexity (lower-${C}^{2}$ 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-${C}^{1}$ 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 A. Lewis [“Robust regularization”, preprint (2002)]. We hereby relate it to the Mifflin semismooth functions.
##### MSC:
 49J52 Nonsmooth analysis (other weak concepts of optimality) 26B25 Convexity and generalizations (several real variables) 47H04 Set-valued operators