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Zonotopes, dicings, and Voronoi’s conjecture on parallelohedra. (English) Zbl 0938.52016
The author settles a special case of Voronoi’s conjecture on parallelohedra, by proving the following theorem: A zonotope which admits a facet-to-facet tiling of Euclidean \(d\)-space by translates is affinely equivalent to the Voronoi polytope of a suitable lattice. He also proves that the Voronoi polytope of a lattice is a zonotope if and only if the corresponding Delaunay tiling is a dicing. The proofs make use of B. A. Venkov’s [Vestnik Leningrad. Univ., Ser. Math. Fiz. Him. 9, 11-31 (1954); see also Usp. Mat. Nauk 9, No. 4(62), 250-251 (1954; Zbl 0056.14103)] and P. McMullen’s [Mathematika 27, 113-121 (1980; Zbl 0432.52016)] characterization of parallelohedra.

MSC:
52C22 Tilings in \(n\) dimensions (aspects of discrete geometry)
52C07 Lattices and convex bodies in \(n\) dimensions (aspects of discrete geometry)
11H31 Lattice packing and covering (number-theoretic aspects)
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[1] Coxeter, H. S.M., The classification of zonohedra by means of projective diagrams, J. Pure Appl. Math., 41, 137-156, (1962) · Zbl 0123.13701
[2] Delaunay, B. N., Sur la partition régulière de l’espace à 4 dimensions, Izv. Akad. Nauk SSSR Otedl. Fiz.-Mat. Nauk., 7, 79-110, (1929) · JFM 56.1120.02
[3] Delaunay, B. A., Piterburgskaja shkola teorii chisel, (1947)
[4] Erdahl, R. M., The theorem of Korkine and zolotarev on the first perfect form, C. R. Math. Rep. Acad. Sci. Canada, 15, 267-272, (1993) · Zbl 0799.52017
[5] Erdahl, R. M.; Ryshkov, S. S., Lattice dicing, C. R. Math. Rep. Acad. Sci. Canada, 15, 255-260, (1993) · Zbl 0817.52019
[6] Erdahl, R. M.; Ryshkov, S. S., On lattice dicing, Europ. J. Combinatorics, 15, 459-481, (1994) · Zbl 0809.52019
[7] Erdahl, R. M.; Ryshkov, S. S., (Behara; Fritsch; Lintz, Recipes for lattice dicing, (1995), Walter de Gruyter New York) · Zbl 0853.52014
[8] C. F. Gauss
[9] Carl Friedrich Gauss Werke, (1876), Göttingen, p. 188-196
[10] J. Reine Angew. Math., 20, 312-320, (1840)
[11] Heller, J., On linear systems with integral valued solutions, Pacific J. Math., 7, 1351-1364, (1957) · Zbl 0079.01903
[12] Korkine, A.; Zolotarev, G., Sur LES formes quadratiques positives, Math. Ann., 11, 242-292, (1877) · JFM 09.0139.01
[13] McMullen, P., On zonotopes, Trans. Am. Math. Soc., 159, 91-110, (1971) · Zbl 0223.52007
[14] McMullen, P., Space tiling zonotopes, Mathematika, 22, 202-211, (1975) · Zbl 0316.52005
[15] McMullen, P., Convex bodies which tile space by translation, Mathematika, 27, 113-121, (1980) · Zbl 0432.52016
[16] Michel, L.; Ryshkov, S. S.; Senechal, M., An extension of voronoi’s theorem on primitive parallelotopes, Europ. J. Combinatorics, 16, 59-63, (1995) · Zbl 0829.52013
[17] Minkowski, H., Allgemeine Lehrsätze über konvexen Polyeder, (1897), p. 198-219 · JFM 28.0427.01
[18] Gesammelte Abhandlungen, 103, 121 · Zbl 0097.30001
[19] Ryshkov, S. S.; Rybnikov, K. A., Generatrissa: the problems of Maxwell and Voronoi, Dokl. Math., 54, 614-617, (1996) · Zbl 0906.51007
[20] Ryshkov, S. S.; Rybnikov, K. A., The theory of quality translation with applications to tilings, Europ. J. Combinatorics, 18, 431-445, (1997) · Zbl 0881.52015
[21] Seeber, L. A., Versuch einer erklärung des innern baues der Körper, Gilberts Ann. Phys., 76, 229-248, (1824)
[22] Seeber, L. A., Untersuchungen über die Eigenschaften der positiven ternären quadratischen Formen, (1831)
[23] Shephard, G. C., Combinatorial properties of associated zonotopes, Can. J. Math., 26, 302-321, (1974) · Zbl 0287.52005
[24] Shephard, G. C., Space filling zonotopes, Mathematika, 21, 169-261, (1974) · Zbl 0296.52004
[25] Venkov, B. A., On a class of euclidean polytopes (in Russian), Vestnik Leningrad. Univ. (Ser. Mat. Fiz. Him.), 9, 11-31, (1954)
[26] Voronoı̈, G., Nouvelles applications des paramètres continus à la théorie des formes quadratiques. premier Mémoire. sur quelques propriètés des formes quadratiques positives parfaites, J. Reine Angew. Math., 133, 79-178, (1908) · JFM 38.0261.01
[27] G. Voronoı̈, Nouvelles applications des paramètres continus à la théorie des formes quadratiques. Deuxième Mémoire. Recherches sur les parallélloèdres primitifs, J. Reine Angew. Math. · JFM 39.0274.01
[28] Partition uniforme de l’espace analytique à n dimensions à l’aide des translations d’un même polyèdre convexe, 134, 198-287, (1908)
[29] Domaines de formes quadratiques correspondant aux différents types de paralléloèdres primitifs, 135, 67-181, (1909) · JFM 40.0267.17
[30] Zhitomirski, O., Verschärfung eines satzes vom woronoi, Z. Leningrad. Fiz.-Mat. Ovsc., 2, 131-151, (1929)
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