Paper information

This paper is a joint paper of Mathieu Dutour-Sikirić, Alexey Garber, and Alexander Magazinov.

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 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 P is combinatorially equivalent to a
Dirichlet-Voronoi polytope for some lattice Λ, and this cominatorial equivalence is naturally translated into equivalence of the
tiling by copies of P with the Voronoi tiling of Λ.

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 and cohomologies of this complex.

The paper is available at arXiv.

The related source code and data files are available under supporting files tab below.

Supporting files

Information about face lattices of five-dimensional Dirichlet-Voronoi parallelohedra is contained in the following files. The parallelohedra are grouped based on the number extreme rays defining the corresponding secondary cone.

- FaceLatt_5_1.txt
- FaceLatt_5_2.txt
- FaceLatt_5_3.txt
- FaceLatt_5_4.txt
- FaceLatt_5_5.txt
- FaceLatt_5_6.txt
- FaceLatt_5_7.txt
- FaceLatt_5_8.txt
- FaceLatt_5_9.txt
- FaceLatt_5_10.txt
- FaceLatt_5_11.txt
- FaceLatt_5_12.txt
- FaceLatt_5_13.txt
- FaceLatt_5_14.txt
- FaceLatt_5_15.txt

- Information about dual 3-cells of five-dimensional Dirichlet-Voronoi parallelohedra is contained in the following files.
The parallelohedra are grouped based on the number extreme rays defining the corresponding secondary cone
- DualCell_data_01.txt
- DualCell_data_02.txt
- DualCell_data_03.txt
- DualCell_data_04.txt
- DualCell_data_05.txt
- DualCell_data_06.txt
- DualCell_data_07.txt
- DualCell_data_08.txt
- DualCell_data_09.txt
- DualCell_data_10.txt
- DualCell_data_11.txt
- DualCell_data_12.txt
- DualCell_data_13.txt
- DualCell_data_14.txt
- DualCell_data_15.txt

- Python scripts that transform face lattice data into dual cell data and generate a GAP program to work with the Venkov complex based on dual cell data.
- SAGE script that checks GGM condition based on dual cell data. You might need to change the filenames in the beginning of the script based on their location.