Covering \(L^p\) spaces by balls (Q472094)

The authors demonstrate that if \(X\) is a separable infinite-dimensional uniformly rotund and uniformly smooth Banach space, then for every covering of \(X\) by closed balls \(B_n = x_n + r_n B_X\), \(r_n > 0\), there is a point \(x \in X\) which belongs to infinitely many \(B_n\). A particular case of this result for \(X\) being a Hilbert space was obtained earlier by \textit{V. P. Fonf} and \textit{C. Zanco} [Can. Math. Bull. 57, No. 1, 42--50 (2014; Zbl 1298.46015)].