Hostname: page-component-8448b6f56d-t5pn6 Total loading time: 0 Render date: 2024-04-25T04:58:19.875Z Has data issue: false hasContentIssue false
  • English
  • Français

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: pradipta@isical.ac.in, sudipta_r@isical.ac.in
S. Dutta
Affiliation:
Department of Mathematics, Ben Gurion University of the Negev, P.O. Box 653, Beer-Sheva, Israel, e-mail: sudipta@cs.bgu.ac.il
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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

[1]Bandyopadhyay, P., ‘The Mazur intersection property for families of closed bounded convex sets in Banach spaces’, Colloq. Math. 63 (1992), 45–56.CrossRefGoogle Scholar
[2]Chen, D. and Lin, B.L., ‘Ball separation properties in Banach spaces’, Rocky Mountain J. Math. 28 (1998), 835–873.CrossRefGoogle Scholar
[3]Deville, R., Godefroy, G. and Zizler, V., Smoothness and renormings in Banach spaces, Pitman Monographs and Surveys in Pure and Applied Mathematics 64 (Longman Scientific & Technical, Harlow, 1993).Google Scholar
[4]Deville, R. and Zizler, V., ‘Farthest points in W ✶-compact sets’, Bull. Austral. Math. Soc. 38 (1988), 433–439.CrossRefGoogle Scholar
[5]Fitzpatrick, S., ‘Metric projections and the differentiability of distance functions’, Bull. Austral. Math. Soc. 22 (1980), 291–312.CrossRefGoogle Scholar
[6]Lau, K.-S., ‘Farthest points in weakly compact sets’, Israel J. Math. 22 (1975), 168–174.CrossRefGoogle Scholar
[7]Preiss, D., ‘Fréchet derivatives of Lipschitz functions’, J. Funct. Anal. 91 (1990), 312–345.CrossRefGoogle Scholar
[8]Westphal, U. and Schwartz, T., ‘Farthest points and monotone operators’, Bull. Austral. Math. Soc. 58 (1998), 75–92.CrossRefGoogle Scholar