On translational clouds for a convex body

For a d-dimensional convex body K let C(K) denote the minimum size of translational clouds for K. That is, C(K) is the minimum number of mutually non-overlapping translates of K which do not overlap K and block all the light rays emanating from any point of K. In this paper we prove the general upper bound C(K)less than or equal to 6(d2+o(d2)). Furthermore, for an arbitrary centrally symmetric d-dimensional convex body S we show C(S)less than or equal to 3(d2+o(d2)). Finally, for the d- dimensional ball B-d we obtain the bounds 2(0.599d2-o(d2)) less than or equal to C(B-d) less than or equal to 2(1.401d2+o(d2)).