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arXiv · 1812.02964

On the Voronoi Conjecture for combinatorially Voronoi parallelohedra in dimension five

Mathieu Dutour Sikirić·Alexey Garber·Alexander Magazinov
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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.

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Mathieu Dutour Sikirić, Alexey Garber, Alexander Magazinov. 2018-12-07. On the Voronoi Conjecture for combinatorially Voronoi parallelohedra in dimension five. https://doi.org/10.1137/18m1235004

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