## Metric spaces with the small ball property

### Volume 148 / 2001

#### Abstract

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.