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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Generalized subdifferential of the distance function


Author: S. Dutta
Journal: Proc. Amer. Math. Soc. 133 (2005), 2949-2955
MSC (2000): Primary 41A65, 41A52; Secondary 46B20
DOI: https://doi.org/10.1090/S0002-9939-05-08153-0
Published electronically: May 9, 2005
MathSciNet review: 2159773
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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.


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Additional Information

S. Dutta
Affiliation: Stat–Math Division, Indian Statistical Institute, 203, B. T. Road, Kolkata 700 108, India
Address at time of publication: Department of Mathematics, Ben-Gurion University of the Negev, P.O.B. 653, Beer-Sheva 84105, Israel
Email: sudipta_r@isical.ac.in, sudipta@math.bgu.ac.il

DOI: https://doi.org/10.1090/S0002-9939-05-08153-0
Keywords: Nearest points, proximinal sets, almost proximinal sets, Chebyshev sets, generalized subdifferential
Received by editor(s): April 10, 2003
Published electronically: May 9, 2005
Communicated by: Jonathan M. Borwein
Article copyright: © Copyright 2005 American Mathematical Society