Parallelohedra defined by quadratic forms
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.
scientific article; zbMATH DE number 6473899
scientific article