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This article is cited in 2 scientific papers (total in 2 papers)
A Class of Affinely Equivalent Voronoi Parallelohedra
A. A. Gavrilyuk Steklov Mathematical Institute of the Russian Academy of Sciences
Abstract:
Given any parallelohedron $P$, its affine class $\mathscr A(P)$, 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 $\mathscr A(P)$ 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.
Keywords:
parallelohedron, Voronoi parallelohedron, affinely equivalent parallelohedra, Venkov graph, Venkov subgraph, orbifold of Voronoi parallelohedra.
Received: 29.09.2011
Citation:
A. A. Gavrilyuk, “A Class of Affinely Equivalent Voronoi Parallelohedra”, Mat. Zametki, 95:5 (2014), 697–707; Math. Notes, 95:5 (2014), 625–633
Linking options:
https://www.mathnet.ru/eng/mzm9257https://doi.org/10.4213/mzm9257 https://www.mathnet.ru/eng/mzm/v95/i5/p697
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Abstract page: | 334 | Full-text PDF : | 153 | References: | 46 | First page: | 25 |
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