On the sum of the Voronoi polytope of a lattice with a zonotope

TL;DR

This paper investigates when the Minkowski sum of a parallelotope and a zonotope results in a parallelotope, providing necessary conditions, classification methods, and geometric descriptions, especially for the root lattice E6.

Contribution

It introduces criteria for the Minkowski sum of a parallelotope and a zonotope to be a parallelotope, including a classification for symmetric lattices and a geometric characterization for E6.

Findings

Identified necessary conditions for P + Z(U) to be a parallelotope.

Developed methods to verify if a given zonotope sum yields a parallelotope.

Classified admissible zonotopes for certain symmetric lattices, including E6.

Abstract

A parallelotope $P$ is a polytope that admits a facet-to-facet tiling of space by translation copies of $P$ along a lattice. The Voronoi cell $P_V(L)$ of a lattice $L$ is an example of a parallelotope. A parallelotope can be uniquely decomposed as the Minkowski sum of a zone closed parallelotope $P$ and a zonotope $Z(U)$, where $U$ is the set of vectors used to generate the zonotope. In this paper we consider the related question: When is the Minkowski sum of a general parallelotope and a zonotope $P+Z(U)$ a parallelotope? We give two necessary conditions and show that the vectors $U$ have to be free. Given a set $U$ of free vectors, we give several methods for checking if $P + Z(U)$ is a parallelotope. Using this we classify such zonotopes for some highly symmetric lattices. In the case of the root lattice $\mathsf{E}_6$, it is possible to give a more geometric description of the admissible sets of vectors $U$. We found that the set of admissible vectors, called free vectors, is described by the well-known configuration of $27$ lines in a cubic. Based on a detailed study of the geometry of $P_V(\mathsf{e}_6)$, we give a simple characterization of the configurations of vectors $U$ such that $P_V(\mathsf{E}_6) + Z(U)$ is a parallelotope. The enumeration yields $10$ maximal families of vectors, which are presented by their description as regular matroids.