Generalized subdifferential of the distance function

Dutta, S.

doi:10.1090/S0002-9939-05-08153-0

Generalized subdifferential of the distance function
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by S. Dutta

Proc. Amer. Math. Soc. 133 (2005), 2949-2955

DOI: https://doi.org/10.1090/S0002-9939-05-08153-0

Published electronically: May 9, 2005

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Abstract:

We derive the proximal normal formula for almost proximinal sets in a smooth and locally uniformly convex Banach space. Our technique leads us to show the generic Fréchet smoothness of the distance function in the case the norm is Fréchet smooth, and we derive a necessary and sufficient condition for the convexity of a Chebyshev set in a Banach space $X$ with norms on $X$ and $X^*$ locally uniformly convex.

References

J. M. Borwein and J. R. Giles, The proximal normal formula in Banach space, Trans. Amer. Math. Soc. 302 (1987), no. 1, 371–381 (English, with French summary). MR 887515, DOI 10.1090/S0002-9947-1987-0887515-5
Frank H. Clarke, Optimization and nonsmooth analysis, Canadian Mathematical Society Series of Monographs and Advanced Texts, John Wiley & Sons, Inc., New York, 1983. A Wiley-Interscience Publication. MR 709590
Robert Deville, Gilles Godefroy, and Václav Zizler, Smoothness and renormings in Banach spaces, Pitman Monographs and Surveys in Pure and Applied Mathematics, vol. 64, Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York, 1993. MR 1211634
Simon Fitzpatrick, Differentiation of real-valued functions and continuity of metric projections, Proc. Amer. Math. Soc. 91 (1984), no. 4, 544–548. MR 746087, DOI 10.1090/S0002-9939-1984-0746087-9
L. P. Vlasov, Almost convex and Čebyšev sets, Mat. Zametki 8 (1970), 545–550 (Russian). MR 276736