Subsmooth sets: Functional characterizations and related concepts

Aussel, D.; Daniilidis, A.; Thibault, L.

doi:10.1090/S0002-9947-04-03718-3

Subsmooth sets: Functional characterizations and related concepts
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by D. Aussel, A. Daniilidis and L. Thibault PDF

Trans. Amer. Math. Soc. 357 (2005), 1275-1301 Request permission

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 $\mbox {dist} (C;\cdot )$, 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 Lewis in 2002. We hereby relate it to the Mifflin semismooth functions.

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