On The Voronoi Conjecture For Combinatorially Voronoi Parallelohedra In Dimension 5

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 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 A, and this combinatorial equivalence is naturally translated into equivalence of the tiling by copies of P with the Voronoi tiling of A. 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.