Subsmooth sets: Functional characterizations and related concepts

D. Aussel, A. Daniilidis, L. Thibault

Research output: Contribution to journalArticleResearchpeer-review

71 Citations (Scopus)

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; ·), 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-C2 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-C1 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. © 2004 American Mathematical Society.
Original languageEnglish
Pages (from-to)1275-1301
JournalTransactions of the American Mathematical Society
Volume357
DOIs
Publication statusPublished - 1 Apr 2005

Keywords

  • Approximately convex functions
  • Submonotone operator
  • Subsmooth sets
  • Variational analysis

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