Local differentiability of distance functions |
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Authors: | R A Poliquin R T Rockafellar L Thibault |
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Institution: | Department of Mathematical Sciences, University of Alberta, Edmonton, Alberta, Canada T6G 2G1 ; Department of Mathematics 354350, University of Washington, Seattle, Washington 98195-4350 ; Laboratoire d'Analyse Convexe, Université Montpellier II, 34095 Montpellier, France |
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Abstract: | Recently Clarke, Stern and Wolenski characterized, in a Hilbert space, the closed subsets for which the distance function is continuously differentiable everywhere on an open ``tube' of uniform thickness around . Here a corresponding local theory is developed for the property of being continuously differentiable outside of on some neighborhood of a point . This is shown to be equivalent to the prox-regularity of at , 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 being locally of class or such that is convex around for some . Prox-regularity of at corresponds further to the normal cone mapping having a hypomonotone truncation around , and leads to a formula for by way of . 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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Keywords: | Variational analysis distance functions single-valued projections proximal normals prox-regularity proximal smoothness primal-lower-nice functions hypomonotone mappings monotone mappings |
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