A Note on Covering by Convex Bodies

[摘要] A classical theorem of Rogers statesthat for any convex body $K$ in $n$-dimensional Euclidean spacethere exists a covering of the space by translates of $K$ withdensity not exceeding $nlog{n}+nloglog{n}+5n$. Rogers' theoremdoes not say anything about the structure of such a covering. Weshow that for sufficiently large values of $n$ the same bound canbe attained by a covering which is the union of $O(log{n})$translates of a lattice arrangement of $K$.

[学科分类] 数学（综合）

[关键词] almost smooth convex body;convex body of constant width;weakly neighbourly antipodal convex polytope;Illumination Conjecture;X-ray number;X-ray Conjecture