Every Banach space with a \(w^{*}\)-separable dual has a \(1+\varepsilon\)-equivalent norm with the ball covering property (Q1044261)

A \textit{countable ball-covering} in a Banach space~\(X\) a is countable collection of balls off the origin such that the union of these balls contains the unit sphere of~\(X\). This concept was introduced by \textit{L.-X.\thinspace Cheng} in [Isr.\ J.\ Math.\ 156, 111--123 (2006; Zbl 1139.46016)]. It easily follows from the Hahn-Banach separation theorem that, if \(X\) admits a countable ball-covering, then \(X^*\) is weak\(^*\)-separable. The authors prove a converse theorem under renorming: if \(X^*\) is weak\(^*\)-separable, then for every \(\varepsilon > 0\) the space \(X\) possesses a \((1+\varepsilon)\)-equivalent norm in which \(X\) admits a countable ball-covering. The same result was obtained independently in [\textit{V.\,P.\thinspace Fonf} and \textit{C.\,Zanco}, Math.\ Ann.\ 344, No.\,4, 939--945 (2009; Zbl 1179.46015)].