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Transactions of the American Mathematical Society

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Local differentiability of distance functions

Authors: R. A. Poliquin, R. T. Rockafellar and L. Thibault
Journal: Trans. Amer. Math. Soc. 352 (2000), 5231-5249
MSC (1991): Primary 49J52, 58C06, 58C20; Secondary 90C30
Published electronically: June 9, 2000
MathSciNet review: 1694378
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Abstract: Recently Clarke, Stern and Wolenski characterized, in a Hilbert space, the closed subsets $C$ for which the distance function $d_{C}$ is continuously differentiable everywhere on an open ``tube'' of uniform thickness around $C$. Here a corresponding local theory is developed for the property of $d_{C}$ being continuously differentiable outside of $C$ on some neighborhood of a point $x\in C$. This is shown to be equivalent to the prox-regularity of $C$ at $x$, which is a condition on normal vectors that is commonly fulfilled in variational analysis and has the advantage of being verifiable by calculation. Additional characterizations are provided in terms of $d_{C}^{2}$ being locally of class $C^{1+}$ or such that $d_{C}^{2}+\sigma \vert\cdot \vert^{2}$ is convex around $x$ for some $\sigma >0$. Prox-regularity of $C$ at $x$ corresponds further to the normal cone mapping $N_{C}$ having a hypomonotone truncation around $x$, and leads to a formula for $P_{C}$ by way of $N_{C}$. The local theory also yields new insights on the global level of the Clarke-Stern-Wolenski results, and on a property of sets introduced by Shapiro, as well as on the concept of sets with positive reach considered by Federer in the finite dimensional setting.

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Additional Information

R. A. Poliquin
Affiliation: Department of Mathematical Sciences, University of Alberta, Edmonton, Alberta, Canada T6G 2G1

R. T. Rockafellar
Affiliation: Department of Mathematics 354350, University of Washington, Seattle, Washington 98195-4350

L. Thibault
Affiliation: Laboratoire d’Analyse Convexe, Université Montpellier II, 34095 Montpellier, France

Keywords: Variational analysis, distance functions, single-valued projections, proximal normals, prox-regularity, proximal smoothness, primal-lower-nice functions, hypomonotone mappings, monotone mappings
Received by editor(s): June 17, 1997
Received by editor(s) in revised form: June 10, 1998
Published electronically: June 9, 2000
Additional Notes: This work was supported in part by the Natural Sciences and Engineering Research Council of Canada under grant OGP41983 for the first author, by the National Science Foundation under grant DMS–9500957 for the second author, and by NATO under grant CRG 960360 for the third author. The authors are grateful for useful discussions with C. Combari, and for helpful comments made by the referee.
Article copyright: © Copyright 2000 American Mathematical Society