On Translational Clouds for a Convex Body

Abstract

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 $C(K) \leqslant 6^{d^2 + o(d^2 )}$. Furthermore, for an arbitrary centrally symmetric d-dimensional convex body S we show $C(S) \leqslant 3^{d^2 + o(d^2 )}$. Finally, for the d-dimensional ball B^d we obtain the bounds $2^{0.599d^2 - o(d^2 )} \leqslant C(B^d ) \leqslant 2^{1.401d^2 + o(d^2 )}$.