Geodesic
Characterizations of Prox-Regular Sets in Uniformly Convex Banach Spaces
Journal of convex analysis, Tome 13 (2006) no. 3, pp. 525-559
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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.
Classification : 49J52, 58C06, 58C20, 90C30
Mots-clés : Distance function, metric projection mapping, uniformly convex Banach space, variational analysis, proximal normal, prox-regular set 
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     author = {F. Bernard and L. Thibault and N. Zlateva},
     title = {Characterizations of {Prox-Regular} {Sets} in {Uniformly} {Convex} {Banach} {Spaces}},
     journal = {Journal of convex analysis},
     pages = {525--559},
     year = {2006},
     volume = {13},
     number = {3},
     url = {http://geodesic.mathdoc.fr/item/JCA_2006_13_3_JCA_2006_13_3_a4/}
}
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F. Bernard; L. Thibault; N. Zlateva. Characterizations of Prox-Regular Sets in Uniformly Convex Banach Spaces. Journal of convex analysis, Tome 13 (2006) no. 3, pp. 525-559. http://geodesic.mathdoc.fr/item/JCA_2006_13_3_JCA_2006_13_3_a4/