This is an announcement for the paper "Covering $L^p$ spaces by balls" by Vladimir P. Fonf, Michael Levin and Clemente Zanco. Abstract: We prove that, given any covering of any separable infinite-dimensional uniformly rotund and uniformly smooth Banach space $X$ by closed balls each of positive radius, some point exists in $X$ which belongs to infinitely many balls. Archive classification: math.FA math.GN Mathematics Subject Classification: Primary 46B20, Secondary 54D20 Submitted from: mlevine@cs.bgu.ac.il The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1212.2817 or http://arXiv.org/abs/1212.2817