This is an announcement for the paper "Thick coverings for the unit ball of a Banach space" by Jesus M. F. Castillo, Pier Luigi Papini and Marilda A. Simoes. Abstract: We study the behaviour of Whitley's thickness constant of a Banach space with respect to $\ell_p$-products and we compute it for classical $L_p$-spaces. Archive classification: math.FA Remarks: This paper is to appear in Houston Journal of Mathematics Submitted from: castillo@unex.es The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1307.4385 or http://arXiv.org/abs/1307.4385