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. A classical theorem of Rogers [3] states that for any convex body K in n-dimensional Euclidean space E there exists a covering of the space by translates of K with density not exceeding n log n + n log log n + 5n. Erdős and Rogers [1] showed that such a covering exists with the additional property that no point is covered by more than e(n log n+n log log n+5n) bodies. Recently, Füredi and Kang [2] used the Local Lemma of Lovász to prove a result only slightly weaker than that of Erdős and Rogers. They showed that for sufficiently large values of n there is a covering of E by translates of any convex body such that each point is covered at most 10n log n times. Neither of these results yields information about the structure of such a covering. Rogers [5] proved the existence of a lattice covering by translates of any convex body in E with density not exceeding n2 log n+O(1) as n →∞. Here we show that with a slight modification of Rogers’ proof of this latter result we can get yet another proof of the bound O(n log n) for the non-lattice case. Moreover, we show that this bound can be reached by a covering which is the union of O(log n) translates of a lattice arrangement. Theorem. For any convex body K in n-dimensional Euclidean space there exists a lattice arrangement of K such that O(log n) translates of this arrangement form a covering of the space with density not exceeding n log n + n log log n + n + o(n) . The proof is based on three lemmas. The first two are modifications of Lemma 3 and Lemma 4 from Rogers’ paper [5], the third one is a direct consequence of a result of W. Schmidt.