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

Bernard, Frédéric; Thibault, Lionel; Zlateva, Nadia

doi:10.1090/S0002-9947-2010-05261-4

Prox-regular sets and epigraphs in uniformly convex Banach spaces: Various regularities and other properties
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by Frédéric Bernard, Lionel Thibault and Nadia Zlateva

Trans. Amer. Math. Soc. 363 (2011), 2211-2247

DOI: https://doi.org/10.1090/S0002-9947-2010-05261-4

Published electronically: November 16, 2010

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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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