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Mat. Zametki, 2014, Volume 95, Issue 5, Pages 697–707 (Mi mz9257)  

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.

Funding Agency Grant Number
Ministry of Education and Science of the Russian Federation 11.G34.31.0053
Russian Foundation for Basic Research 11-01-00633-а


DOI: https://doi.org/10.4213/mzm9257

Full text: PDF file (511 kB)
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English version:
Mathematical Notes, 2014, 95:5, 625–633

Bibliographic databases:

Document Type: Article
UDC: 514.174+514.87
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

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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. Garber A., Gavrilyuk A., Magazinov A., “The Voronoi Conjecture For Parallelohedra With Simply Connected Delta-Surfaces”, Discret. Comput. Geom., 53:2 (2015), 245–260  crossref  mathscinet  zmath  isi  scopus
    2. Sikiric M.D., Garber A., Schuermann A., Waldmann C., “The complete classification of five-dimensional Dirichlet–Voronoi polyhedra of translational lattices”, Acta Crystallogr. Sect. A, 72:6 (2016), 673–683  crossref  mathscinet  zmath  isi  scopus
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