A Note on Covering by Convex Bodies

A classical theorem of Rogers states that for any convex body K in n-dimensional Euclidean space there exists a covering of the space by translates of K with density not exceeding n log n + n log log n + 5n. Rogers' theorem does not say anything about the structure of such a covering. We show that for sufficiently large values of n the same bound can be attained by a covering which is the union of O(log n) translates of a lattice arrangement of K.