Geodesic
Parallelohedra defined by quadratic forms
Informatics and Automation, Geometry, topology, and applications, Tome 288 (2015), pp. 95-108.

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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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V. P. Grishukhin. Parallelohedra defined by quadratic forms. Informatics and Automation, Geometry, topology, and applications, Tome 288 (2015), pp. 95-108. http://geodesic.mathdoc.fr/item/TRSPY_2015_288_a5/

[1] Voronoï G., “Nouvelles applications des paramètres continus à la théorie des formes quadratiques. Deuxième mémoire: Recherches sur les paralléloèdres primitifs”, J. reine angew. Math., 134 (1908), 198–287 ; 136 (1909), 67–178 | Zbl | Zbl

[2] Venkov B.A., “Ob odnom klasse evklidovykh mnogogrannikov”, Vestn. Leningr. un-ta. Matematika. Fizika. Khimiya, 1954, no. 2, 11–31 | MR

[3] McMullen P., “Convex bodies which tile space by translation”, Mathematika, 27 (1980), 113–121 | DOI | MR | Zbl

[4] Delone B.N., Peterburgskaya shkola teorii chisel, Izd-vo AN SSSR, M.; L., 1947

[5] Dolbilin N.P., “Teoremy Minkovskogo o paralleloedrakh i ikh obobscheniya”, UMN, 62:4 (2007), 157–158 | DOI | MR | Zbl

[6] Dolbilin N.P., “Svoistva granei paralleloedrov”, Tr. MIAN, 266, 2009, 112–126 | MR | Zbl

[7] Grishukhin V.P., “Mnogogranniki Delone i Voronogo kornevoi reshetki $E_7$ i dvoistvennoi reshetki $E_7^*$”, Tr. MIAN, 275, 2011, 68–86 | MR | Zbl

[8] Grishukhin V.P., “A definition of type domain of a parallelotope”, Model. i analiz inform. sistem, 20:6 (2013), 129–134 | MR

[9] Conway J.H., Sloane N.J.A., Sphere packings, lattices and groups, Grundl. Math. Wiss., 290, Springer, Berlin, 1988 ; Konvei Dzh., Sloen N., Upakovki sharov, reshetki i gruppy, Mir, M., 1990 | MR | Zbl

[10] Aigner M., Kombinatornaya teoriya, Mir, M., 1982 | MR

[11] Erdahl R.M., “Zonotopes, dicings, and Voronoi's conjecture on parallelohedra”, Eur. J. Comb., 20 (1999), 527–549 | DOI | MR | Zbl

[12] Ryshkov S.S., “Stroenie $n$-mernogo paralleloedra pervogo tipa”, DAN SSSR, 146:5 (1962), 1027–1030 | Zbl

[13] Bolshakova E.A., “Paralleloedry pervogo tipa i ikh simvoly”, Chebyshev. sb., 7:2 (2006), 38–65 | MR | Zbl

[14] Emelichev V.A., Kovalev M.M., Kravtsov M.K., Mnogogranniki, grafy, optimizatsiya, Nauka, M., 1981 | MR

[15] Ryshkov S.S., Bolshakova E.A., “K teorii korennykh paralleloedrov”, Izv. RAN. Ser. mat., 69:6 (2005), 187–210 | DOI | MR | Zbl

[16] Björner A., Las Vergnas M., Sturmfels B., Wite N., Ziegler G., Oriented matroids, Encycl. Math. Appl., 46, Cambridge Univ. Press, Cambridge, 1999 | MR | Zbl

[17] Garber A., Poyarkov A., “On permutahedra // Voronoï's impact on modern science”, Proc. 3rd Voronoï Conference on Analytic Number Theory and Spatial Tessellations, Book 3, Inst. Math., Kyiv, 2005, 137–145 | Zbl