Voronoi's Conjecture for extensions of Voronoi parallelohedra

TL;DR

This paper proves that if Voronoi's Conjecture holds for a parallelohedron P, then it also holds for the Minkowski sum of P with a segment I, extending the class of parallelohedra satisfying the conjecture.

Contribution

It establishes that Voronoi's Conjecture is preserved under Minkowski addition with segments, broadening the understanding of Voronoi parallelohedra.

Findings

Voronoi's Conjecture holds for P+I if it holds for P.

The Minkowski sum with a segment preserves the Voronoi property.

Extension of Voronoi's Conjecture to a larger class of parallelohedra.

Abstract

Let $I$ be a segment in the $d$-dimensional Euclidean space $\mathbb E^d$. Let $P$ and $P+I$ be parallelohedra in $\mathbb E^d$, where "+" denotes the Minkowski sum. We prove that Voronoi's Conjecture holds for $P+I$, i.e. $P+I$ is a Voronoi parallelohedron for some Euclidean metric in $\mathbb E^d$, if Voronoi's Conjecture holds for $P$.