Minimal ball-coverings in Banach spaces and their application
Volume 192 / 2009
Abstract
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.
