Parallelohedra: A retrospective and new results

Dolbilin, N.

doi:10.1090/S0077-1554-2013-00208-3

Parallelohedra: A retrospective and new results
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by N. P. Dolbilin
Translated by: E. Khukhro

Trans. Moscow Math. Soc. 2012, 207-220

DOI: https://doi.org/10.1090/S0077-1554-2013-00208-3

Published electronically: March 21, 2013

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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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