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Transactions of the American Mathematical Society

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Subsmooth sets: Functional characterizations and related concepts

Authors: D. Aussel, A. Daniilidis and L. Thibault
Journal: Trans. Amer. Math. Soc. 357 (2005), 1275-1301
MSC (2000): Primary 26B25; Secondary 49J52, 47H04
Published electronically: November 23, 2004
MathSciNet review: 2115366
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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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Additional Information

D. Aussel
Affiliation: Département de Mathématiques, Université de Perpignan, 66860 Perpignan Cedex, France

A. Daniilidis
Affiliation: Departament de Matemàtiques, Universitat Autònoma de Barcelona, E-08193 Bellaterra (Cerdanyola del Vallès), Spain

L. Thibault
Affiliation: Université Montpellier II, Département de Mathématiques, CC 051, Place Eugène Bataillon, 34095 Montpellier Cedex 05, France

Keywords: Variational analysis, subsmooth sets, submonotone operator, approximately convex functions
Received by editor(s): February 24, 2003
Published electronically: November 23, 2004
Additional Notes: The research of the second author has been supported by the Spanish Ministry of Education Program: “Ayudas para estancias de profesores e investigadores extranjeros en España” (Grant No SB2000-0369).
Article copyright: © Copyright 2004 American Mathematical Society

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