Name
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 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. |
Year | DOI | Venue |
|---|---|---|
2020 | 10.1137/18M1235004 | SIAM JOURNAL ON DISCRETE MATHEMATICS |
Keywords | DocType | Volume |
tiling, parallelohedron, Voronoi conjecture | Journal | 34 |
Issue | ISSN | Citations |
4 | 0895-4801 | 0 |
PageRank | References | Authors |
0.34 | 0 | 3 |
Name | Order | Citations | PageRank |
|---|---|---|---|
| Mathieu Dutour Sikiric | 1 | 18 | 4.50 |
| A. Garber | 2 | 1 | 1.72 |
| Alexander Magazinov | 3 | 1 | 2.85 |