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Zbl 1094.49016
Aussel, D.; Daniilidis, A.; Thibault, L.
Subsmooth sets: functional characterizations and related concepts.
(English)
[J] Trans. Am. Math. Soc. 357, No. 4, 1275-1301 (2005). ISSN 0002-9947; ISSN 1088-6850/e

Summary: Prox-regularity of a set [{\it R. A. Poliquin, R. T. Rockafellar} and {\it L. Thibault}, Trans. Am. Math. Soc. 352, No. 11, 5231--5249 (2000; Zbl 0960.49018)], or its global version, proximal smoothness [{\it F. H. Clarke, R. J. Stern} and {\it 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 $\text{dist}\,(C;\cdot)$, 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 {\it A. Lewis} [Robust regularization'', preprint (2002)]. We hereby relate it to the Mifflin semismooth functions.
MSC 2000:
*49J52 Nonsmooth analysis (other weak concepts of optimality)
26B25 Convexity and generalizations (several real variables)
47H04 Set-valued operators

Keywords: subsmooth sets; submonotone operators; approximately convex functions

Citations: Zbl 0881.49008; Zbl 0960.49018

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