Geodesic
Layer Superposition of the Root Lattice~$A_n$
Matematičeskie zametki, Tome 109 (2021) no. 2, pp. 219-234.

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An explicit description of a Dirichlet–Voronoi (DV) cell of the root lattice $A_n$ in the form of the convex hull of an $n$-dimensional parallelotope with a vertex at 0 and its copies centrally symmetric with respect to 0 is given. Remarkable properties of this DV-cell are described, which manifest themselves when layers of the lattice $A_n$ are superposed. It is shown that the one-parameter family of their metric forms runs through the cone of positivity from one boundary to the other, passing through four $L$-type domains.
Keywords: Dirichlet–Voronoi cell, root lattice $A_n$, superposition of layers. 
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V. P. Grishukhin. Layer Superposition of the Root Lattice~$A_n$. Matematičeskie zametki, Tome 109 (2021) no. 2, pp. 219-234. http://geodesic.mathdoc.fr/item/MZM_2021_109_2_a5/

[1] G. F. Voronoi, “Issledovanie o primitivnykh paralleloedrakh”, Sobranie sochinenii, T. 2, Izd-vo AN USSR, Kiev, 1952, 239–368

[2] Dzh. Konvei, N. Sloen, Upakovki sharov, reshetki i gruppy, Mir, M., 1990 | MR

[3] B. Delaunay, “Sur la partition régulière de l'espace à 4 dimensions. Première partie”, Izvestiya Akademii nauk SSSR. VII seriya. Otdelenie fiziko-matematicheskikh nauk, 1929, no. 1, 79–110 | Zbl

[4] B. N. Delone, “Geometriya polozhitelnykh kvadratichnykh form. Chast II”, UMN, 1938, no. 4, 102–164

[5] E. S. Barnes, “The construction of perfect and extreme forms. II”, Acta Arith., 5 (1959), 205–222 | DOI | MR

[6] E. P. Baranovskii, “Razbienie evklidovykh prostranstv na $L$-mnogogranniki nekotorykh sovershennykh reshetok”, Diskretnaya geometriya i topologiya, Tr. MIAN SSSR, 196, Nauka, M., 1991, 27–46 | MR | Zbl

[7] V. P. Grishukhin, “Svobodnye i nesvobodnye mnogogranniki Voronogo”, Matem. zametki, 80:3 (2006), 367–378 | DOI | MR | Zbl

[8] B. A. Venkov, “O proektirovanii paralleloedrov”, Matem. sb., 49 (91):2 (1959), 207–224 | Zbl

[9] V. P. Grishukhin, “Paralleloedry, opredelyaemye kvadratichnymi formami”, Geometriya, topologiya i prilozheniya, Tr. MIAN, 288, MAIK «Nauka/Interperiodika», M., 2015, 95–108 | DOI

[10] N. P. Dolbilin, “Svoistva granei paralleloedrov”, Geometriya, topologiya i matematicheskaya fizika. II, Tr. MIAN, 266, MAIK «Nauka/Interperiodika», M., 2009, 112–126 | MR | Zbl

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

[12] P. McMullen, “Space tiling zonotopes”, Mathematika, 22 (1975), 202–211 | DOI | MR

[13] F. Jäger, “On space tiling zonotopes and regular chain groups”, Ars Combin., 16-B (1983), 257–270 | MR

[14] R. M. Erdahl, “Zonotopes, dicings, and Voronoi's conjecture on parallelohedra”, European J. Combin., 20:6 (1999), 527–549 | DOI | MR

[15] D. Sikirić, V .Grishukhun, “Zonotopes and parallelotopes”, Southeast Asian Bull. Math., 41:2 (2017), 197–207 | MR

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

[17] J. H. Conway, N. J. A. Sloane, “The cell structures of certain lattices”, Miscellanea Mathematica, Springer-Verlag, Berlin, 1991, 71–107 | MR

[18] V. P. Grishukhin, Pervaya i vtoraya sovershennye oblasti polozhitelnykh kvadratichnykh form, Preprint TsEMI RAN No320, M., 2016

[19] V. P. Grishukhin, “The Delaunay tilings of a continuous family of lattices”, Symmetry: Culture and Science, 22:1-2 (2011), 27–54