On the Voronoi Conjecture for Combinatorially Voronoi Parallelohedra in Dimension 5

Authors

Mathieu Dutour Sikiric
Alexey Garber, The University of Texas Rio Grande Valley
Alexander Magazinov

Document Type

Article

Publication Date

2020

Abstract

In a recent paper, Garber, Gavrilyuk, and Magazinov [Discrete Comput. Geom., 53 (2015), pp. 245--260] proposed a sufficient combinatorial condition for a parallelohedron to be affinely Voronoi. We show that this condition holds for all 5-dimensional Voronoi parallelohedra. Consequently, the Voronoi conjecture in $\mathbb{R}^5$ holds if and only if every 5-dimensional parallelohedron is combinatorially Voronoi. Here, by saying that a parallelohedron $P$ is combinatorially Voronoi, we mean that $P$ is combinatorially equivalent to a Dirichlet--Voronoi polytope for some lattice $\Lambda$, and this combinatorial equivalence is naturally translated into equivalence of the tiling by copies of $P$ with the Voronoi tiling of $\Lambda$. We also propose a new condition which, if satisfied by a parallelohedron $P$, is sufficient to infer that $P$ is affinely Voronoi. The condition is based on the new notion of the Venkov complex associated with a parallelohedron and cohomologies of this complex.

Comments

© 2020, Society for Industrial and Applied Mathematics
Read More: https://epubs.siam.org/doi/abs/10.1137/18M1235004?journalCode=sjdmec

Recommended Citation

Sikirić, Mathieu Dutour, Alexey Garber, and Alexander Magazinov. 2020. “On the Voronoi Conjecture for Combinatorially Voronoi Parallelohedra in Dimension 5.” SIAM Journal on Discrete Mathematics, 2481–2501. https://doi.org/10.1137/18M1235004.

Publication Title

SIAM Journal on Discrete Mathematics

DOI

10.1137/18M1235004