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 Tr. Mat. Inst. Steklova, 2015, Volume 288, Pages 95–108 (Mi tm3602)

Parallelohedra defined by quadratic forms

V. P. Grishukhin

Central Economics and Mathematics Institute, RAS, Moscow, Russia

Abstract: The results of Section III of G. F. Voronoi's famous memoir are presented in modern terms. The description of a parallelohedron by a system of linear constraints with quadratic right-hand side naturally leads to the notion of a contact face, which is called a standard face by N. P. Dolbilin. It is proved that a nonempty intersection of two contact faces generates a $4$- or $6$-belt of these contact faces. As an example, zonotopes defined by quadratic forms are considered. In particular, zonotopal parallelohedra of Voronoi's principal domain are examined in detail. It is shown that these parallelohedra are submodular polytopes, which are frequently encountered in combinatorial theory.

DOI: https://doi.org/10.1134/S0371968515010069

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English version:
Proceedings of the Steklov Institute of Mathematics, 2015, 288, 81–93

Bibliographic databases:

UDC: 511.9
Received in August 2014

Citation: V. P. Grishukhin, “Parallelohedra defined by quadratic forms”, Geometry, topology, and applications, Collected papers. Dedicated to Professor Nikolai Petrovich Dolbilin on the occasion of his 70th birthday, Tr. Mat. Inst. Steklova, 288, MAIK Nauka/Interperiodica, Moscow, 2015, 95–108; Proc. Steklov Inst. Math., 288 (2015), 81–93

Citation in format AMSBIB
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