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Farthest points and the farthest distance map

Published online by Cambridge University Press:  17 April 2009

Pradipta Bandyopadhyay  and
S. Dutta
Pradipta Bandyopadhyay
Affiliation:
Stat-Math Division, Indian Statistical Institute, 203, B. T. Road, Kolkata 700 108, India, e-mail: [email protected], [email protected]
S. Dutta
Affiliation:
Department of Mathematics, Ben Gurion University of the Negev, P.O. Box 653, Beer-Sheva, Israel, e-mail: [email protected]
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In this paper, we consider farthest points and the farthest distance map of a closed bounded set in a Banach space. We show, inter alia, that a strictly convex Banach space has the Mazur intersection property for weakly compact sets if and only if every such set is the closed convex hull of its farthest points, and recapture a classical result of Lau in a broader set-up. We obtain an expression for the subdifferential of the farthest distance map in the spirit of Preiss' Theorem which in turn extends a result of Westphal and Schwartz, showing that the subdifferential of the farthest distance map is the unique maximal monotone extension of a densely defined monotone operator involving the duality map and the farthest point map.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 71 , Issue 3 , June 2005 , pp. 425 - 433
Copyright
Copyright © Australian Mathematical Society 2005

References

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