Minimal ball-coverings in Banach spaces and their application

Lixin 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 -