On the density of sets avoiding parallelohedron distance 1

Christine Bachoc 1 Thomas Bellitto 2, 3 Philippe Moustrou 1 Arnaud Pêcher 3, 2
3 Realopt - Reformulations based algorithms for Combinatorial Optimization
LaBRI - Laboratoire Bordelais de Recherche en Informatique, IMB - Institut de Mathématiques de Bordeaux, Inria Bordeaux - Sud-Ouest
Abstract : 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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https://hal.archives-ouvertes.fr/hal-01567118
Contributor : Christine Bachoc <>
Submitted on : Thursday, July 27, 2017 - 5:39:56 PM
Last modification on : Tuesday, April 17, 2018 - 9:04:45 AM

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  • HAL Id : hal-01567118, version 1
  • ARXIV : 1708.00291

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Christine Bachoc, Thomas Bellitto, Philippe Moustrou, Arnaud Pêcher. On the density of sets avoiding parallelohedron distance 1. 2017. ⟨hal-01567118⟩

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