Journal of Convex Analysis 13 (2006), No. 3, 525--559
Copyright Heldermann Verlag 2006
Characterizations of Prox-Regular 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.univ-montp2.fr
Lionel Thibault
Dép. de Mathématiques, Université Montpellier II, CC 051, Place Eugène Bataillon, 34095 Montpellier Cedex 5, France
thibault@math.univ-montp2.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 prox-regular sets in Hilbert spaces. Prox-regularity 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 prox-regularity of a closed set C at x is shown to be still equivalent to the property of the distance function dC 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 prox-regularity corresponding to the continuous differentiability of the distance function dC 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, prox-regular set.
MSC: 49J52, 58C06, 58C20; 90C30
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