# Minimal ball-coverings in Banach spaces and their application

Lixin Cheng; Qingjin Cheng; Huihua Shi

Studia Mathematica (2009)

- Volume: 192, Issue: 1, page 15-27
- ISSN: 0039-3223

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topLixin Cheng, Qingjin Cheng, and Huihua Shi. "Minimal ball-coverings in Banach spaces and their application." Studia Mathematica 192.1 (2009): 15-27. <http://eudml.org/doc/286362>.

@article{LixinCheng2009,

abstract = {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.},

author = {Lixin Cheng, Qingjin Cheng, Huihua Shi},

journal = {Studia Mathematica},

keywords = {ball-covering; reflexivity; smoothness; differentiability; Banach space},

language = {eng},

number = {1},

pages = {15-27},

title = {Minimal ball-coverings in Banach spaces and their application},

url = {http://eudml.org/doc/286362},

volume = {192},

year = {2009},

}

TY - JOUR

AU - Lixin Cheng

AU - Qingjin Cheng

AU - Huihua Shi

TI - Minimal ball-coverings in Banach spaces and their application

JO - Studia Mathematica

PY - 2009

VL - 192

IS - 1

SP - 15

EP - 27

AB - 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.

LA - eng

KW - ball-covering; reflexivity; smoothness; differentiability; Banach space

UR - http://eudml.org/doc/286362

ER -

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