The seven dimensional perfect Delaunay polytopes and Delaunay simplices

TL;DR

This paper classifies perfect Delaunay polytopes in dimension 7, introduces an enumeration algorithm based on the Erdahl cone, and identifies only two such polytopes, providing insights into Delaunay simplices in this dimension.

Contribution

It presents a new algorithm for enumerating perfect Delaunay polytopes using the Erdahl cone and applies it to dimension 7, discovering only two such polytopes and classifying Delaunay simplices.

Findings

Only two perfect Delaunay polytopes in dimension 7: $3_{21}$ and Erdahl Rybnikov polytope.

Identified 11 types of Delaunay simplices in dimension 7.

Algorithm for enumeration of perfect Delaunay polytopes based on the Erdahl cone.

Abstract

For a lattice $L$ of $R^n$, a sphere $S(c,r)$ of center $c$ and radius $r$ is called {\em empty} if for any $v\in L$ we have $\Vert v - c\Vert \geq r$. Then the set $S(c,r)\cap L$ is the vertex set of a {\em Delaunay polytope} $P=conv(S(c,r)\cap L)$. A Delaunay polytope is called {\em perfect} if any affine transformation $\phi$ such that $\phi(P)$ is a Delaunay polytope is necessarily an isometry of the space composed with an homothety. Perfect Delaunay polytopes are remarkable structure that exist only if $n=1$ or $n\geq 6$ and they have shown up recently in covering maxima studies. Here we give a general algorithm for their enumeration that relies on the Erdahl cone. We apply this algorithm in dimension 7 which allow us to find that there are only two perfect Delaunay polytopes: $3_{21}$ which is a Delaunay polytope in the root lattice $\mathsf{E}_7$ and the Erdahl Rybnikov polytope. We then use this classification in order to get the list of all types Delaunay simplices in dimension 7 and found 11 types.