Parallelohedra defined by quadratic forms
type: Item
instance of: scholarly article
title: Parallelohedra defined by quadratic forms
zbMATH Open document ID: 1326.52018
DOI: 10.1134/S008154381501006X
published in: Proceedings of the Steklov Institute of Mathematics
publication date: 2015-08-20T00:00:00Z
review text: The author presents the results of Section III, which is titled ''Determination of parallelohedra with the help of positive quadratic forms,'' of \textit{G. Voronoï}'s famous memoir [J. Reine Angew. Math. 134, 198--287 (1908; JFM 39.0274.01)] and [ibid. 136, 67--178 (1909; JFM 40.0267.17)] in modern terms. Among the most interesting contributions of the paper under review we mention a new proof of the fact that the polytope \(P(a)=\{x\in \mathbb R^n:\langle p, x\rangle \leq a(p), p\in\mathcal P\}\), where \(\langle p, x\rangle\) is the scalar product of vectors \(p,q\in \mathbb R^n\), \(a\) is a quadratic form, and \(\mathcal P\) is the set of so-called normal vectors, is a parallelohedron if and only if the following two conditions are satisfied: (1) \(P\) and all its facets are centrally symmetric and (2) \(P\) has only 4- and 6-belts. The author calls the conditions (1) and (2) Venkov's conditions because they appeared in [\textit{B.A. Venkov}, ``On a class of Euclidean polytopes'', Vestnik Leningrad. Univ., Ser. Math. Fiz. Him. 9, 11--31 (1954); see also Usp. Mat. Nauk 9, No.4, 250--251 (1954; Zbl 0056.14103)] for the first time and independently in [\textit{P. McMullen}, Mathematika 27, 113--121 (1980; Zbl 0432.52016)]. For zonotopes defined by quadratic forms, the author studies in detail the Voronoi's principal domain and shows that these parallelohedra are submodular polytopes.
reviewed by: Victor Alexandrov
Mathematics Subject Classification ID: 52C22
Mathematics Subject Classification ID: 52B12
Mathematics Subject Classification ID: 11H06
zbMATH DE Number: 6473899
zbMATH Keywords: parallelohedron
zbMATH Keywords: polytope
zbMATH Keywords: scalar product
zbMATH Keywords: quadratic form
zbMATH Keywords: Voronoi cell
zbMATH Keywords: Venkov's conditions
author: Viacheslav Grishukhin
MaRDI profile type: Publication
cites work: Convex bodies which tile space by translation
cites work: Q5789047
cites work: Minkowski's theorems on parallelohedra and their generalisations
cites work: Properties of faces of parallelohedra
cites work: Delaunay and Voronoi polytopes of the root lattice \(E_{7}\) and of the dual lattice \(E^\ast_{7}\)
cites work: Q4039784
cites work: Q3050437
cites work: Zonotopes, dicings, and Voronoi's conjecture on parallelohedra
cites work: Q5531624
cites work: Q3573598
cites work: Q3673576
cites work: On the theory of mainstay parallelohedra
cites work: Oriented Matroids
cites work: Q5430528
description: scientific article; zbMATH DE number 6473899