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Inscribed centers, reflexivity, and some applications

Published online by Cambridge University Press:  09 April 2009

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.

Type
Research Article
Copyright
Copyright © Australian Mathematical Society 1986

References

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