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Prox-regular sets and epigraphs in uniformly convex Banach spaces: Various regularities and other properties


Authors: Frédéric Bernard, Lionel Thibault and Nadia Zlateva
Journal: Trans. Amer. Math. Soc. 363 (2011), 2211-2247
MSC (2010): Primary 49J52, 58C06, 58C20; Secondary 90C30
DOI: https://doi.org/10.1090/S0002-9947-2010-05261-4
Published electronically: November 16, 2010
MathSciNet review: 2746681
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Abstract: We continue the study of prox-regular sets that we began in a previous work in the setting of uniformly convex Banach spaces endowed with a norm both uniformly smooth and uniformly convex (e.g., $ L^p, W^{m,p}$ spaces). We prove normal and tangential regularity properties for these sets, and in particular the equality between Mordukhovich and proximal normal cones. We also compare in this setting the proximal normal cone with different Hölderian normal cones depending on the power types $ s,q$ of moduli of smoothness and convexity of the norm. In the case of sets that are epigraphs of functions, we show that $ J$-primal lower regular functions have prox-regular epigraphs and we compare these functions with Poliquin's primal lower nice functions depending on the power types $ s,q$ of the moduli. The preservation of prox-regularity of the intersection of finitely many sets and of the inverse image is obtained under a calmness assumption. A conical derivative formula for the metric projection mapping of prox-regular sets is also established. Among other results of the paper it is proved that the Attouch-Wets convergence preserves the uniform $ r$-prox-regularity property and that the metric projection mapping is in some sense continuous with respect to this convergence for such sets.


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Additional Information

Frédéric Bernard
Affiliation: Département de Mathématiques, Université Montpellier II, CC 051, Place Eugène Bataillon, 34095 Montpellier cedex 5, France
Email: bernard@math.univ-montp2.fr

Lionel Thibault
Affiliation: Département de Mathématiques, Université Montpellier II, CC 051, Place Eugène Bataillon, 34095 Montpellier cedex 5, France
Email: thibault@math.univ-montp2.fr

Nadia Zlateva
Affiliation: Department of Mathematics and Informatics, Sofia University, 5 James Bourchier blvd., 1164 Sofia, Bulgaria
Email: zlateva@fmi.uni-sofia.bg

DOI: https://doi.org/10.1090/S0002-9947-2010-05261-4
Keywords: Distance function, metric projection mapping, uniformly convex Banach space, variational analysis, proximal normal, prox-regular set, epigraph
Received by editor(s): October 22, 2008
Received by editor(s) in revised form: November 9, 2009
Published electronically: November 16, 2010
Additional Notes: The third author was partially supported by the Bulgarian National Fund for Scientific Research, contract DO 02-360/2008.
Article copyright: © Copyright 2010 American Mathematical Society

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