Metric spaces with the small ball property

Ehrhard Behrends, and Vladimir M. Kadets. "Metric spaces with the small ball property." Studia Mathematica 148.3 (2001): 275-287. <http://eudml.org/doc/286233>.

@article{EhrhardBehrends2001,
abstract = {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.},
author = {Ehrhard Behrends, Vladimir M. Kadets},
journal = {Studia Mathematica},
keywords = {small ball property; compact set},
language = {eng},
number = {3},
pages = {275-287},
title = {Metric spaces with the small ball property},
url = {http://eudml.org/doc/286233},
volume = {148},
year = {2001},
}

TY - JOUR
AU - Ehrhard Behrends
AU - Vladimir M. Kadets
TI - Metric spaces with the small ball property
JO - Studia Mathematica
PY - 2001
VL - 148
IS - 3
SP - 275
EP - 287
AB - 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.
LA - eng
KW - small ball property; compact set
UR - http://eudml.org/doc/286233
ER -