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.