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New metric properties for prox-regular sets

Authors :
Florent Nacry
Samir Adly
Lionel Thibault
Mathématiques & Sécurité de l'information (XLIM-MATHIS)
XLIM (XLIM)
Université de Limoges (UNILIM)-Centre National de la Recherche Scientifique (CNRS)-Université de Limoges (UNILIM)-Centre National de la Recherche Scientifique (CNRS)
LAboratoire de Mathématiques et PhySique (LAMPS)
Université de Perpignan Via Domitia (UPVD)
Institut Montpelliérain Alexander Grothendieck (IMAG)
Université de Montpellier (UM)-Centre National de la Recherche Scientifique (CNRS)
Source :
Mathematical Programming
Publication Year :
2020
Publisher :
HAL CCSD, 2020.

Abstract

In this paper, we present diverse new metric properties that prox-regular sets shared with convex ones. At the heart of our work lie the Legendre-Fenchel transform and complements of balls. First, we show that a connected prox-regular set is completely determined by the Legendre-Fenchel transform of a suitable perturbation of its indicator function. Then, we prove that such a function is also the right tool to extend, to the context of prox-regular sets, the famous connection between the distance function and the support function of a convex set. On the other hand, given a prox-regular set, we examine the intersection of complements of open balls containing the set. We establish that the distance of a point to a prox-regular set is the maximum of the distances of the point from boundaries of all such complements separating the set and the point. This is in the line of the known result expressing the distance from a convex set in terms of separating hyperplanes. To the best of our knowledge, these results are new in the literature and show that the class of prox-regular sets have good properties known in convex analysis. Mathematics Subject Classification (2010) 49J52 · 49J53

Details

Language :
English
Database :
OpenAIRE
Journal :
Mathematical Programming
Accession number :
edsair.doi.dedup.....026e73705f534633fdd7cd51d5f147a9