The strong and uniform ball-covering properties (Q2660487)

A family of balls \(\{B(x_n,r_n)\}\) of a normed space \(X\) is a strong ball-covering family if there exists a positive number \(r>0\) such that \(S_X\subset \bigcup B(x_n,r_n)\), \(\|x_n\|>r_n\) for any \(n\in\mathbb{N}\), and \(\sup r_n\leq r\); in this case it is said that \(X\) has the strong ball-covering property (SBCP). If in addition there exists a positive number \(\delta >0\) such that \(B(x_n,r_n)\cap \delta B_X=\emptyset\) for every \(n\in\mathbb{N}\), then the family \(\{B(x_n,r_n)\}\) is called uniformly ball-covering and \(X\) has the uniform ball-covering property (UBCP). The authors show that these two notions differ. In Example~2.3 it is shown that an \(\ell_{\infty}\)-sum of a countable family of spaces \(\{X_{\lambda_k}\}\), each defined by a renorming of \(\ell_{\infty}\) depending on the real number \(\lambda_k\), has SBCP but lacks UBCP. The rest of the paper is devoted to several stability results. Among them, \(X\) has the UBCP if and only if \(Y\) has the UBCP, whenever \(Y\) is a dense subset of \(X\) (Lemma~3.2). If \(X\) has the SBCP, then \(L_{p}([0,1],X)\) has the SBCP, too, whenever \(1\leq p<\infty\) (Theorem 3.6), while for the same values of \(p\), \(X\) has the UBCP if and only if \(L_{p}([0,1],X)\) has the UBCP (Theorem~3.7).