On the Density of Sets Avoiding Parallelohedron Distance 1

The maximal density of a measurable subset of ℝ^n avoiding Euclidean distance 1 is unknown except in the trivial case of dimension 1. In this paper, we consider the case of a distance associated to a polytope that tiles space, where it is likely that the sets avoiding distance 1 are of maximal density 2^-n , as conjectured by Bachoc and Robins. We prove that this is true for n=2 , and for the Voronoi regions of the lattices A_n , n≥ 2 .