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Parallelohedra: A retrospective and new results


Author: N. P. Dolbilin
Translated by: E. Khukhro
Original publication: Trudy Moskovskogo Matematicheskogo Obshchestva.
Journal: Trans. Moscow Math. Soc. 2012, 207-220
MSC (2010): Primary 52B20; Secondary 52B11, 52B12
DOI: https://doi.org/10.1090/S0077-1554-2013-00208-3
Published electronically: March 21, 2013
MathSciNet review: 3184976
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Abstract: Parallelohedra are polyhedra that partition Euclidean space with parallel copies. This class of polyhedra has applications both in mathematics and in the natural sciences. An important subclass of parallelohedra is comprised of the so-called Voronoĭ parallelohedra, which are Dirichlet-Voronoĭ domains for integer lattices. More than a century ago Voronoĭ stated the conjecture that every parallelohedron is affinely equivalent to some Voronoĭ parallelohedron. Although considerable progress has been made, this conjecture has not been proved in full. This paper contains a survey of a number of classical theorems in the theory of parallelohedra, together with some new results related to Voronoĭ's conjecture.


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Additional Information

N. P. Dolbilin
Affiliation: Steklov Mathematical Institute of the Russian Academy of Sciences
Email: dolbilin@mi.ras.ru

DOI: https://doi.org/10.1090/S0077-1554-2013-00208-3
Keywords: Parallelohedron, Minkowski--Venkov criterion for parallelohedra, Vorono\u{\i}'s conjecture on parallelohedra, standard face, index of a face
Published electronically: March 21, 2013
Additional Notes: This research was supported by the Government of the Russian Federation (grant no. 11.G34.31.0053) and the Russian Foundation for Basic Research (grant no. 11-01-00633-a).
Article copyright: © Copyright 2013 American Mathematical Society