
Journal of Convex Analysis 13 (2006), No. 3, 525559 Copyright Heldermann Verlag 2006 Characterizations of ProxRegular Sets in Uniformly Convex Banach Spaces Frédéric Bernard Dép. de Mathématiques, Université Montpellier II, CC 051, Place Eugène Bataillon, 34095 Montpellier Cedex 5, France bernard@math.univmontp2.fr Lionel Thibault Dép. de Mathématiques, Université Montpellier II, CC 051, Place Eugène Bataillon, 34095 Montpellier Cedex 5, France thibault@math.univmontp2.fr Nadia Zlateva Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, Acad. G. Bonchev Str. Bl. 8, 1113 Sofia, Bulgaria zlateva@math.bas.bg The aim of the paper is to extend to the setting of uniformly convex Banach spaces the results obtained for proxregular sets in Hilbert spaces. Proxregularity of a set C at a point x of C is a variational condition related to normal vectors and which is common to many types of sets. In the context of uniformly convex Banach spaces, the proxregularity of a closed set C at x is shown to be still equivalent to the property of the distance function d_{C} to be continuously differentiable outside of C on some neighbourhood of x. Additional characterizations are provided in terms of metric projection mapping. We also examine the global level of proxregularity corresponding to the continuous differentiability of the distance function d_{C} over an open tube of uniform thickness around the set C. Keywords: Distance function, metric projection mapping, uniformly convex Banach space, variational analysis, proximal normal, proxregular set. MSC: 49J52, 58C06, 58C20; 90C30 [ Fulltextpdf (926 KB)] for subscribers only. 