# On the Voronoi Conjecture for combinatorially Voronoi parallelohedra in dimension five

## TL;DR

This paper proves that all five-dimensional Voronoi parallelohedra satisfy a key combinatorial condition, advancing the understanding of the Voronoi conjecture in five dimensions and introducing a new criterion based on the Venkov complex.

## Contribution

The paper demonstrates that the combinatorial condition for a parallelohedron to be affinely Voronoi holds in five dimensions, and introduces a new sufficient condition involving the Venkov complex.

## Key findings

- The combinatorial condition holds for all five-dimensional Voronoi parallelohedra.
- The Voronoi conjecture in $\
- A new sufficient condition for a parallelohedron to be affinely Voronoi is proposed.

## Abstract

In a recent paper Garber, Gavrilyuk and Magazinov proposed a sufficient combinatorial condition for a parallelohedron to be affinely Voronoi. We show that this condition holds for all five-dimensional Voronoi parallelohedra. Consequently, the Voronoi conjecture in $\mathbb R^5$ holds if and only if every five-dimensional parallelohedron is combinatorially Voronoi. Here, by saying that a parallelohedron $P$ is combinatorially Voronoi, we mean that the tiling $\mathcal T(P)$ by translates of $P$ is combinatorially isomorphic to some tiling $\mathcal T(P')$, where $P'$ is a Voronoi parallelohedron, and that the isomorphism naturally induces a linear isomorphism of lattices $\Lambda(P)$ and $\Lambda(P')$. We also propose a new sufficient condition implying that a parallelohedron is affinely Voronoi. The condition is based on the new notion of the Venkov complex associated with a parallelohedron.