Metric spaces with the small ball property

Ehrhard Behrends¹ ; Vladimir M. Kadets²

¹ I. Mathematisches Institut Freie Universität Berlin Arnimallee 2–6 D-14 195 Berlin, Germany
² Department of Mechanics and Mathematics Kharkov National University 4 Svobody Sq. Kharkov, 61077 Ukraine

Studia Mathematica, Tome 148 (2001) no. 3, pp. 275-287

Résumé

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.

Détail

DOI : 10.4064/sm148-3-6

Keywords: metric space said have small ball property sbp every varepsilon possible write union sequence closed balls smaller varepsilon mathop lim study permanence properties examples sbp main results paper following nbsp bounded convex closed sets banach spaces have sbp only compact nbsp precisely finite dimensional banach spaces have sbp generally complete metric group has sbp separable locally compact nbsp boundary bidual infinite dimensional banach space does have sbp particular set extreme points unit ball infinite dimensional reflexive banach space fails have sbp