Metric spaces with the small ball property
A metric space (M;d) is said to have the small ball property (sbp) if for every "0 > 0 it is possible to write M as the union of a sequence (B(xn;rn)) of closed balls such that the rn are smaller than "0 and limrn = 0. We study permanence properties and examples of sbp. The main results of this paper are the following: 1. Bounded convex closed sets in Banach spaces have sbp only if they are compact. 2. Precisely the nite- dimensional Banach spaces have sbp. (More generally: a complete metric group has sbp i it is separable and locally compact.) 3. Let B be a boundary in the bidual of an innite-dimension al Banach space. Then B does not have sbp. In particular the set of extreme points in the unit ball of an innite-dimension al reexiv e Banach space fails to have sbp.