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Voronoĭ's conjecture and space tiling zonotopes

Part of: Polytopes and polyhedra

Published online by Cambridge University Press:  26 February 2010

Michel Deza  and
Viacheslav Grishukhin
Michel Deza
Affiliation:
Ecole Normale Supérieure, Paris, France. E-mail: Michel.Deza@ens.fr
Viacheslav Grishukhin
Affiliation:
CEMI, Russian Academy of Sciences, Moscow, Russia. E-mail: grishuhn@cemi.rssi.ru
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Abstract

Voronoĭ conjectured that every parallelotope is affinely equivalent to a Voronoĭ polytope. For some m, a parallelotope is defined by a set of m facet vectors pi, and defines a set of m lattice vectors ti, for 1≤i≤m. It is shown that Voronoĭ's conjecture is true for an n-dimensional parallelotope P if and only if there exist scalars γi, and a positive definite n × n matrix Q such that γipi = Qti for each i. In this case, the quadratic form f(x) = xTQx is the metric form of P.

MSC classification

Secondary: 52B22: Shellability

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Type
Research Article
Information
Mathematika , Volume 51 , Issue 1-2 , December 2004 , pp. 1 - 10
Copyright
Copyright © University College London 2004

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