By a ball-covering of a Banach space X, we mean a collection of open balls off the origin in X and whose union contains the unit sphere of X; a ball-covering is called minimal if its cardinality is smallest among all ball-coverings of X. This article, through establishing a characterization for existence of a ball-covering in Banach spaces, shows that for every n ∈ ℕ with k ≤ n there exists an n-dimensional space admitting a minimal ball-covering of n + k balls. As an application, we give a new characterization of superreflexive spaces in terms of ball-coverings. Finally, we show that every infinite-dimensional Banach space admits an equivalent norm such that there is an infinite-dimensional quotient space possessing a countable ball-covering.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-sm192-1-2,
author = {Lixin Cheng and Qingjin Cheng and Huihua Shi},
title = {Minimal ball-coverings in Banach spaces and their application},
journal = {Studia Mathematica},
volume = {192},
year = {2009},
pages = {15-27},
zbl = {1176.46015},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm192-1-2}
}
Lixin Cheng; Qingjin Cheng; Huihua Shi. Minimal ball-coverings in Banach spaces and their application. Studia Mathematica, Tome 192 (2009) pp. 15-27. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm192-1-2/