The Star Number of Coverings of Space with Convex Bodies

1. When a system of sets covers a space, the star number of the covering is the supremum over the sets of the system of the cardinals of the numbers of sets of the system meeting a set of the system . The standard Lebesgue 'brick-laying' construction provides an example, for each positive integer n, of a lattice covering of R" by closed rectangular parallelepipeds with star number 2"T'-1 . In view of the results of dimension theory, it is natural to conjecture that any covering of R" by closed sets of uniformly bounded diameter has star number at least 2"-1 ; and this has been proved by V . Boltyanskii [1] in the special case n = 2 . In this paper we consider only coverings of R" by translates of a fixed convex body. We first give a simple proof (the idea of which comes from the work of Minkowski and Voronoi) of THEOREM 1 . The star number of a lattice covering of R" by translates of a closed symmetrical convex body is at least 2` 1 -1 . Then we consider the problem of constructing coverings of R" by translates of a given closed convex body K with as small a star number as possible . By a minor modification of method we used in [2] we prove THEOREM 2. Provided n is sufficiently large, if K is a closed convex body in Rn with difference body DK, there is a covering of R" by translates of K with star number less than