Delaunay polytopes derived from the Leech lattice

TL;DR

This paper explores Delaunay polytopes derived from the Leech lattice, discovering new perfect polytopes with unique properties, disproving longstanding conjectures, and establishing bounds on lattice covering radii.

Contribution

It introduces novel perfect Delaunay polytopes in Leech lattice-derived lattices, including those with larger lattice widths and special symmetry properties, and provides bounds on covering radii.

Findings

Found perfect Delaunay polytopes with lattice width 4

Disproved several longstanding conjectures about Delaunay polytopes

Established an upper bound for the covering radius of certain lattices

Abstract

Given a lattice L of R^n, a polytope D is called a Delaunay polytope in L if the set of its vertices is S\cap L where S is a sphere having no lattice points in its interior. D is called perfect if the only ellipsoid in R^n that contains S\cap L is exactly S. For a vector v of the Leech lattice \Lambda_{24} we define \Lambda_{24}(v) to be the lattice of vectors of \Lambda_{24} orthogonal to v. We studied Delaunay polytopes of L=\Lambda_{24}(v) for |v|^2<=22. We found some remarkable examples of Delaunay polytopes in such lattices and disproved a number of long standing conjectures. In particular, we discovered: --Perfect Delaunay polytopes of lattice width 4; previously, the largest known width was 2. --Perfect Delaunay polytopes in L, which can be extended to perfect Delaunay polytopes in superlattices of L of the same dimension. --Polytopes that are perfect Delaunay with respect to two lattices $L\subset L'$ of the same dimension. --Perfect Delaunay polytopes D for L with |Aut L|=6|Aut D|: all previously known examples had |Aut L|=|Aut D| or |Aut L|=2|Aut D|. --Antisymmetric perfect Delaunay polytopes in L, which cannot be extended to perfect (n+1)-dimensional centrally symmetric Delaunay polytopes. --Lattices, which have several orbits of non-isometric perfect Delaunay polytopes. Finally, we derived an upper bound for the covering radius of \Lambda_{24}(v)^{*}, which generalizes the Smith bound and we prove that it is met only by \Lambda_{23}^{*}, the best known lattice covering in R^{23}.