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 $\tau$ 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 $\tau$. In the present paper we investigate how small the dimension of this compact set can be, in the case the members of $\tau$ 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 $\tau$ are rotund or smooth bodies in the usual sense.
Coverings of Banach spaces: beyond the Corson theorem / V.P. Fonf, C. Zanco. - In: FORUM MATHEMATICUM. - ISSN 0933-7741. - 21:3(2009), pp. 533-546.
Coverings of Banach spaces: beyond the Corson theorem
C. ZancoUltimo
2009
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 $\tau$ 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 $\tau$. In the present paper we investigate how small the dimension of this compact set can be, in the case the members of $\tau$ 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 $\tau$ are rotund or smooth bodies in the usual sense.Pubblicazioni consigliate
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