Abstract

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∈Cx \in 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 dCd_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 prox-regularity corresponding to the continuous differentiability of the distance function dCd_C over an open tube of uniform thickness around the set C.

Author information

Contact details are reproduced from the original publication and may be historical.

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

Keywords

  • Distance function
  • metric projection mapping
  • uniformly convex Banach space
  • variational analysis
  • proximal normal
  • prox-regular set

Mathematics Subject Classification

49J52, 58C06, 58C20; 90C30

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Suggested citation

F. Bernard, L. Thibault, N. Zlateva. “Characterizations of Prox-Regular Sets in Uniformly Convex Banach Spaces.” Journal of Convex Analysis 13 (2006), No. 3/4, 525–559. https://doi.org/10.68381/jca13044

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Published by Heldermann Verlag, 2006. Rights now held by Banach Press LLC.