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Coverings of Banach spaces: beyond the Corson theorem

Coverings of Banach spaces: beyond the Corson theorem A well-known result due to H. Corson has been recently improved by the authors. In its final form it essentially reads as follows: for any covering τ by closed bounded convex subsets of any Banach space X containing a separable infinite-dimensional dual space, a (algebraically) finite-dimensional compact set C can always be found that meets infinitely many members of τ . In the present paper we investigate how small the dimension of this compact set can be, in the case the members of τ are closed bounded convex bodies satisfying general conditions of rotundity or smoothness type. In particular, such a compact set turns out to be a segment whenever the members of τ are rotund or smooth bodies in the usual sense. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Forum Mathematicum de Gruyter

Coverings of Banach spaces: beyond the Corson theorem

Forum Mathematicum , Volume 21 (3) – May 1, 2009

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References (10)

Publisher
de Gruyter
Copyright
© de Gruyter 2009
ISSN
0933-7741
eISSN
1435-5337
DOI
10.1515/FORUM.2009.026
Publisher site
See Article on Publisher Site

Abstract

A well-known result due to H. Corson has been recently improved by the authors. In its final form it essentially reads as follows: for any covering τ by closed bounded convex subsets of any Banach space X containing a separable infinite-dimensional dual space, a (algebraically) finite-dimensional compact set C can always be found that meets infinitely many members of τ . In the present paper we investigate how small the dimension of this compact set can be, in the case the members of τ are closed bounded convex bodies satisfying general conditions of rotundity or smoothness type. In particular, such a compact set turns out to be a segment whenever the members of τ are rotund or smooth bodies in the usual sense.

Journal

Forum Mathematicumde Gruyter

Published: May 1, 2009

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