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.
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References
G. Voronoï, “Nouvelles applications des paramètres continus à la théorie des formes quadratiques. Deuxième mémoire: Recherches sur les paralléloèdres primitifs,” J. Reine Angew. Math. 134, 198–287 (1908); 136, 67–178 (1909).
B. A. Venkov, “On a class of Euclidean polytopes,” Vestn. Leningrad. Univ., Ser. Mat. Fiz. Khim., No. 2, 11–31 (1954).
P. McMullen, “Convex bodies which tile space by translation,” Mathematika 27, 113–121 (1980).
B. N. Delone, The St.-Petersburg School of Number Theory (Akad. Nauk SSSR, Moscow, 1947; Am. Math. Soc., Providence, RI, 2005).
N. P. Dolbilin, “Minkowski’s theorems on parallelohedra and their generalisations,” Usp. Mat. Nauk 62 (4), 157–158 (2007) [Russ. Math. Surv. 62, 793–795 (2007)].
N. P. Dolbilin, “Properties of faces of parallelohedra,” Tr. Mat. Inst. im. V.A. Steklova, Ross. Akad. Nauk 266, 112–126 (2009) [Proc. Steklov Inst. Math. 266, 105–119 (2009)].
V. P. Grishukhin, “Delaunay and Voronoi polytopes of the root lattice E 7 and of the dual lattice E 7*,” Tr. Mat. Inst. im. V.A. Steklova, Ross. Akad. Nauk 275, 68–86 (2011) [Proc. Steklov Inst. Math. 275, 60–77 (2011)].
V. P. Grishukhin, “A definition of type domain of a parallelotope,” Model. Anal. Inf. Sist. 20 (6), 129–134 (2013).
J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups (Springer, Berlin, 1988), Grundl. Math. Wiss. 290.
M. Aigner, Combinatorial Theory (Springer, Berlin, 1979).
R. M. Erdahl, “Zonotopes, dicings, and Voronoi’s conjecture on parallelohedra,” Eur. J. Comb. 20, 527–549 (1999).
S. S. Ryshkov, “The structure of an n-dimensional parallelohedron of the first kind,” Dokl. Akad. Nauk SSSR 146 (5), 1027–1030 (1962) [Sov. Math., Dokl. 3, 1451–1454 (1962)].
E. A. Bol’shakova, “Parallelohedra of the first kind and their symbols,” Chebyshev. Sb. 7 (2), 38–65 (2006).
V. A. Emelichev, M. M. Kovalev, and M. K. Kravtsov, Polytopes, Graphs and Optimisation (Nauka, Moscow, 1981; Cambridge Univ. Press, Cambridge, 1984).
S. S. Ryshkov and E. A. Bol’shakova, “On the theory of mainstay parallelohedra,” Izv. Ross. Akad. Nauk, Ser. Mat. 69 (6), 187–210 (2005) [Izv. Math. 69, 1257–1277 (2005)].
A. Björner, M. Las Vergnas, B. Sturmfels, N. Wite, and G. Ziegler, Oriented Matroids (Cambridge Univ. Press, Cambridge, 1999), Encycl. Math. Appl. 46.
A. Garber and A. Poyarkov, “On permutahedra,” in Voronoï’s Impact on Modern Science: Proc. 3rd Voronoï Conference on Analytic Number Theory and Spatial Tessellations (Inst. Math., Kyiv, 2005), Book 3, pp. 137–145.
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Grishukhin, V.P. Parallelohedra defined by quadratic forms. Proc. Steklov Inst. Math. 288, 81–93 (2015). https://doi.org/10.1134/S008154381501006X
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DOI: https://doi.org/10.1134/S008154381501006X