Metric spaces with the small ball property

Volume 148 / 2001

Ehrhard Behrends, Vladimir M. Kadets Studia Mathematica 148 (2001), 275-287 MSC: 46B10, 46B20, 54E35. DOI: 10.4064/sm148-3-6

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.

Authors

  • Ehrhard BehrendsI. Mathematisches Institut
    Freie Universität Berlin
    Arnimallee 2–6
    D-14 195 Berlin, Germany
    e-mail
  • Vladimir M. KadetsDepartment of Mechanics and Mathematics
    Kharkov National University
    4 Svobody Sq.
    Kharkov, 61077 Ukraine
    e-mail

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