A class of affinely equivalent Voronoi parallelohedra
A class of affinely equivalent Voronoi parallelohedra
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一类仿射等价 Voronoi 平行六面体
DOI：
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发表时间：
2014
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通讯作者：
A. A. Gavrilyuk
中科院分区：
文献类型：
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作者：
A. A. Gavrilyuk
给定任何平行六面体P，其仿射类A（P），即，所有平行六面体仿射等价于它的集合被考虑。这个仿射类是否包含至少一个Voronoi平行面体，即，一个平行面体，它是一个Dirichlet域的一些格？这个问题，通常被称为Voronoi猜想，一百多年来一直没有答案。证明了当A（P）中的Voronoi平行面体的子集非空时，该子集是一个orbifold，并且它的维数（作为一个具有奇点的真实的流形）完全由它的组合类型决定，即它等于给定平行面体的所谓Venkov子图的连通分支数.然而，这个轨道折叠的结构不仅取决于平行面体的组合性质，还取决于其仿射性质。
Given any parallelohedron P, its affine class A (P), i.e., the set of all parallelohedra affinely equivalent to it, is considered. Does this affine class contain at least one Voronoi parallelohedron, i.e., a parallelohedron which is a Dirichlet domain for some lattice? This question, more commonly known as Voronoi’s conjecture, has remained unanswered for more than a hundred years. It is shown that, in the case where the subset of Voronoi parallelohedra in A (P) is nonempty, this subset is an orbifold, and its dimension (as a real manifold with singularities) is completely determined by its combinatorial type; namely, it is equal to the number of connected components of the so-called Venkov subgraph of the given parallelohedron. Nevertheless, the structure of this orbifold depends not only on the combinatorial properties of the parallelohedron but also on its affine properties.