Minimal ball-coverings in Banach spaces and their application

Lixin Cheng¹ ; Qingjin Cheng¹ ; Huihua Shi¹

¹Department of Mathematics Xiamen University Xiamen 361005, China

Studia Mathematica, Tome 192 (2009) no. 1, pp. 15-27

Résumé

By a ball-covering $\mathcal{B}$ 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 $\mathcal{B}$ is called minimal if its cardinality $\mathcal{B}^\#$ 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 $n\in \mathbb{N}$ with $k\leq n$ 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.

Détail

DOI : 10.4064/sm192-1-2

Keywords: ball covering mathcal banach space mean collection balls off origin whose union contains unit sphere nbsp ball covering mathcal called minimal its cardinality mathcal smallest among ball coverings nbsp article through establishing characterization existence ball covering banach spaces shows every mathbb leq there exists n dimensional space admitting minimal ball covering balls application characterization superreflexive spaces terms ball coverings finally every infinite dimensional banach space admits equivalent norm there infinite dimensional quotient space possessing countable ball covering