On the density of sets avoiding parallelohedron distance 1

TL;DR

This paper investigates the maximal density of sets in Euclidean space avoiding a specific polytope-based distance, proving the conjectured density for certain lattice regions in dimensions two and higher.

Contribution

It proves the conjecture that the maximal density is 2^-n for sets avoiding a polytope-based distance in specific lattice regions, extending known results.

Findings

Confirmed the conjecture for n=2.

Extended results to Voronoi regions of lattices An for n>=2.

Established maximal density as 2^-n in these cases.

Abstract

The maximal density of a measurable subset of R^n 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 Vorono\"i regions of the lattices An, n >= 2.