Ball-Covering Property in Uniformly Non- l 3 ( 1 ) Banach Spaces and Application

Abstract

This paper shows the following. (1) $X$ is a uniformly non-$l_3^{(1)}$ space if and only if there exist two constants $\alpha ,\beta >0$ such that, for every 3-dimensional subspace $Y$ of $X$, there exists a ball-covering $𝔅$ of $Y$ with $c(𝔅)=\mathrm{4}$ or $5$ which is $\alpha$-off the origin and $r(𝔅)\le \beta$. (2) If a separable space $X$ has the Radon-Nikodym property, then $X^{\text{*}}$ has the ball-covering property. Using this general result, we find sufficient conditions in order that an Orlicz function space has the ball-covering property.