On the density of sets avoiding parallelohedron distance 1

The maximal density of a measurable subset of $${{\mathbb {R}}}^n$$ 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 $$A_n$$ , $$n\ge 2$$ .