On the density of sets avoiding parallelohedron distance 1
Résumé
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 Voronoï regions of the lattices An, n >= 2.
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