On the density of sets avoiding parallelohedron distance 1

2017/01/01 by Bachoc, Christine, Bellitto, Thomas, Moustrou, Philippe +5
Materials Science · Mathematics · #Quasicrystal Structures and Properties #Limits and Structures in Graph Theory #Mathematical Approximation and Integration

paper · pdf · doi:10.48550/arxiv.1708.00291

Abstract

The maximal density of a measurable subset of Rⁿ avoiding Euclidean distance1 is unknown except in the trivial case of dimension 1. In this paper, we consider thecase of a distance associated to a polytope that tiles space, where it is likely that the setsavoiding 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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• The density of sets avoiding distance 1 in Euclidean space
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• Distance sets corresponding to convex bodies
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