Lattice covering of space with symmetric convex bodies

In 1959 C. A. Rogers gave the following estimate for the density vL(K) of lattice-coverings of euclidean d-space Edwith convex bodies K:Vl(K)≤dlog2loged+cHere, c is a suitable constant which does not depend on D and K. Moreover, Rogers proved that for the unit ball Bdthe upper bound can be replaced by cd(loged)(1/2)log22πe, which is, of course a major improvement. In the present paper we show that such an improvement can be obtained for a larger class of convex bodies. In particular, we prove the following theorem. Let K be a convex body in Ed, and let k be an integer satisfying k > log2loged + 4. If there exist at least k hyperplanes Hl,…, Hkwith normals mutually perpendicular and an affine transformation A such that A(K) is symmetrical with respect to Hl…, Hk, respectively, then Vl(K)≤cd(loged)1+log2 e.Actually, for a bound of this type we do not even need any symmetry assumption. In fact, some weaker properties concerning shadow boundaries will suffice.