Preprint No.
A-00-23
Ehrhard Behrends, Vladimir Kadets
Metric spaces with the small ball property
Abstract:
A metric space $(M,d)$ is said to have {\em the small
ball property (sbp)} if for every $\eps_{0}>0$ it is possible to write
$M$ as a union of a sequence $(B(x_{n},r_{n}))$ of closed balls
such that the $r_{n}$
are smaller than $\eps_{0}$ and $\lim r_{n}=0$.
We study permanence properties and examples, the main results of this
paper are the following: 1. Bounded convex closed sets in Banach
spaces have the {\em sbp} only if they are compact.
2. Precisely the finite dimensional Banach spaces have the {\em sbp}.
(More generally:
a complete metric group has the {\em 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 the {\em sbp}. In particular the
set of extreme points in the unit ball of
infinite dimensional reflexive Banach spaces
fails to have the {\em sbp}.
Keywords: metric space, precompact, Banach space, extreme point,
reflexive Banach space
Mathematics Subject Classification (MSC91): 46B10, 46B20, 54E35
Language: ENG
Available: Pr-A-00-23.ps
Contact: Ehrhard Behrends, Freie Universität Berlin, Fachbereich Mathematik und Informatik, Arnimallee 2-6, D-14195 Berlin, Germany (behrends@math.fu-berlin.de)
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