A metric space (M,d) is said to have the small ball property (sbp) if for every ε₀ > 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 ε₀ and lim rₙ = 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.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-sm148-3-6,
author = {Ehrhard Behrends and Vladimir M. Kadets},
title = {Metric spaces with the small ball property},
journal = {Studia Mathematica},
volume = {147},
year = {2001},
pages = {275-287},
zbl = {1002.46012},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm148-3-6}
}
Ehrhard Behrends; Vladimir M. Kadets. Metric spaces with the small ball property. Studia Mathematica, Tome 147 (2001) pp. 275-287. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm148-3-6/