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Inscribed centers, reflexivity, and some applications

Part of: Normed linear spaces and Banach spaces; Banach lattices Nonlinear operators and their properties

Published online by Cambridge University Press:  09 April 2009

A. A. Astaneh
A. A. Astaneh
Affiliation:
Department of Mathematics, University of Mashhad, Mashhad, Iran
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Abstract

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We first define an inscribed center of a bounded convex body in a normed linear space as the center of a largest open ball contained in it (when such a ball exists). We then show that completeness is a necessary condition for a normed linear space to admit inscribed centers. We show that every weakly compact convex body in a Banach space has at least one inscribed center, and that admitting inscribed centers is a necessary and sufficient condition for reflexivity. We finally apply the concept of inscribed center to prove a type of fixed point theorem and also deduce a proposition concerning so-called Klee caverns in Hilbert spaces.

MSC classification

Secondary: 46B20: Geometry and structure of normed linear spaces 46B10: Duality and reflexivity 47H10: Fixed-point theorems
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 41 , Issue 3 , December 1986 , pp. 317 - 324
Copyright
Copyright © Australian Mathematical Society 1986

References

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[2]Astaneh, A. A., “Completeness of normed linear spaces admitting centers”, J. Austral. Math. Soc. Ser. A 39 (1985) 360–366.Google Scholar
[3]Dunford, N. and Schwartz, J. T., Linear Operators, Vol. I (Interscience, New York, 1958).Google Scholar
[4]James, R. C., “Characterizations of reflexivity”, Studia Math. 23 (1964), 205–216.Google Scholar
[5]Jameson, G. J. O., Topology and normed spaces (Chapman and Hall, London, 1974).Google Scholar