SIAM Journal on Discrete Mathematics, Volume 34, Issue 4, Page 2481-2501, January 2020.
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.

中文翻译：

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关于维数为5的Voronoi Parallelohedra的Voronoi猜想

SIAM离散数学杂志，第34卷，第4期，第2481-2501页，2020年1月。
在最近的一篇论文中，Garber，Gavrilyuk和Magazinov [离散计算。Geom。，53（2015），pp。245--260]提出了平行六面体仿射Voronoi的充分组合条件。我们证明此条件适用于所有5维Voronoi平行面体。因此，当且仅当每个5维平行六面体组合为Voronoi时，$ \ mathbb {R} ^ 5 $中的Voronoi猜想成立。在这里，通过说平行六面体$ P $在组合上是Voronoi，我们的意思是$ P $在某种晶格$ \ Lambda $上组合等效于Dirichlet-Voronoi多面体，并且这种组合对等自然地转换为平铺的等效由$ P $的副本和Voronoi平铺$ \ Lambda $组成。我们还提出了一个新条件，如果满足平行六面体$ P $，足以推断$ P $是Voronoi的仿射。该条件基于与平行六面体和该复合体的同调性相关联的Venkov复合体的新概念。