Journal of Convex Analysis 30 (2023), No. 4, 1217--1240
Copyright Heldermann Verlag 2023
Farthest Distance Function to Strongly Convex Sets
Florent Nacry
Lab. de Modélisation Pluridisciplinaire et Simulations, Université de Perpignan, France, Perpignan, France
florent.nacry@univ-perp.fr
Vo Anh Thuong Nguyen
Lab. de Modélisation Pluridisciplinaire et Simulations, Université de Perpignan, France
vo-anh-thuong.nguyen@univ-perp.fr
Lionel Thibault
Inst. Montpelliérain A. Grothendieck, Université de Montpellier, France, France
and: Centro de Modelamiento Matematico, Universidad de Chile, Santiago, Chile
lionel.thibault@umontpellier.fr
The aim of the present paper is twofold. On one hand, we show that the strong convexity of a set is equivalent to the semiconcavity of its associated farthest distance function. On the other hand, we establish that the farthest distance of a point from a strongly convex set is the minimum of farthest distances of the given point from suitable closed balls separating the set and the point. Various other results on strongly convex sets are also provided.
Keywords: Variational analysis, strong convexity, prox-regularity, farthest distance function, semiconvexity.
MSC: 49J52, 49J53.
[ Fulltext-pdf (183 KB)] for subscribers only.