On the Voronoi Conjecture for combinatorially Voronoi parallelohedra in dimension five

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 $\mathcal{T}(P^{\prime })$, where $P^{\prime }$ is a Voronoi parallelohedron, and that the isomorphism naturally induces a linear isomorphism of lattices $\Lambda (P)$ and $\Lambda (P^{\prime })$. 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.