Metric spaces with the small ball property
Tom 148 / 2001
Streszczenie
A metric space $(M,d)$ is said to have the small ball property (sbp) if for every $\varepsilon _{0}>0$ it is possible to write $M$ as the union of a sequence $(B(x_{n},r_{n}))$ of closed balls such that the $r_{n}$ are smaller than $\varepsilon _{0}$ and $\mathop {\rm lim}r_{n}=0$. 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 finite-dimensional Banach spaces have sbp. (More generally: a complete metric group has sbp iff it is separable and locally compact.) 3. Let $B$ be a boundary in the bidual of an infinite-dimensional Banach space. Then $B$ does not have sbp. In particular the set of extreme points in the unit ball of an infinite-dimensional reflexive Banach space fails to have sbp.
