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Voronoĭ's conjecture and space tiling zonotopes

Part of: Polytopes and polyhedra

Published online by Cambridge University Press:  26 February 2010

Michel Deza  and
Viacheslav Grishukhin
Michel Deza
Affiliation:
Ecole Normale Supérieure, Paris, France. E-mail: Michel.Deza@ens.fr
Viacheslav Grishukhin
Affiliation:
CEMI, Russian Academy of Sciences, Moscow, Russia. E-mail: grishuhn@cemi.rssi.ru
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Abstract

Voronoĭ conjectured that every parallelotope is affinely equivalent to a Voronoĭ polytope. For some m, a parallelotope is defined by a set of m facet vectors pi, and defines a set of m lattice vectors ti, for 1≤i≤m. It is shown that Voronoĭ's conjecture is true for an n-dimensional parallelotope P if and only if there exist scalars γi, and a positive definite n × n matrix Q such that γipi = Qti for each i. In this case, the quadratic form f(x) = xTQx is the metric form of P.

MSC classification

Secondary: 52B22: Shellability
Type
Research Article
Information
Mathematika , Volume 51 , Issue 1-2 , December 2004 , pp. 1 - 10
Copyright
Copyright © University College London 2004

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References

[Aig79]Aigner, M.. Combinatorial Theory (Grundlehren der mathematischen Wissenshaften 234). Springer Verlag (Berlin, Heidelberg, New York, 1979).Google Scholar
[A154]Aleksandrov, A. D.. On filling of space by polytopes (in Russian). Vestnik Leningradskogo Univ., Ser. math. phys. chem., 9 (1954), 33–43.Google Scholar
[B-Z92]Bjőrner, A.Las Vergnas, M.Sturmfels, B.White, N. and Ziegler, G. M.. Oriented Matroids (Encyclopedia of Mathematics and its Applications 46). Cambridge Univ. Press (1999).CrossRefGoogle Scholar
[Cox62]Coxeter, H. S. M.. The classification of zonohedra by means of projective diagrams. J. Math. Pure Appl. (9), 41 (1962), 137–156.Google Scholar
[De29]Delaunay, B. N.. Sur la partition régulierè de l'espace á 4 dimensions. Izvestia AN SSSR. oldel. phys.-math. nauk, 1 (1929), 79–110; 2 (1929), 145–164.Google Scholar
[En98]Engel, P.. Investigations of parallelohedra in ℝd. Voronoĭs impact on Modern Science (ed. Engel, P. and Syta, H.), Institute of Mathematics, vol. 2 (Kyiv, 1998), 22–60.Google Scholar
[En00]Engel, P.. The contraction types of parallelohedra in E5. Ada Cryst., Sect. A 56 (2000), 491–496.Google ScholarPubMed
[EG02]Engel, P. and Grishukhin, V.. There are exactly 222 L-types of primitive 5-dimensional parallelotopes. Europ. J. Combinatories (to appear).Google Scholar
[Er99]Erdahl, R. M.. Zonotopes, dicings and Voronoĭ's conjecture on parallelohedra. Europ. J. Combinatorics, 20 (1999), 527–549.CrossRefGoogle Scholar
[ErRy94]Erdahl, R. M. and Ryshkov, S. S.. On lattice dicing. Europ. J. Combinatorics, 15 (1994), 459481.CrossRefGoogle Scholar
[Jae83]Jaeger, F.. On space-tiling zonotopes and regular chain groups. Ars Combinatoria, 16-B (1983). 257–270.Google Scholar
[McM75]McMullen, P.. Space tiling zonotopes. Mathematika, 22 (1975), 202–211.CrossRefGoogle Scholar
[McM80]McMullen, P.. Convex bodies which tile space by translation. Mathematika, 27 (1980), 113–121.CrossRefGoogle Scholar
[RB76]Ryshkov, S. S. and Baranovskii, E. P.. C-types of n-dimensional lattices and 5-dimensional primitive parallelohedra (with application to the theory of coverings). Trudy Math. Inst. im. Steklova, 137 (1976) (transl. as Proc. Steklov hist. Math. 4 (1978))Google Scholar
[Sht73]Shtogrin, M. I.. Regular Dirichlet-Voronoĭ partitions for the second triclinic group. Proc. Steklov Inst. Math., 123 (1973), 1–127.Google Scholar
[Sh75]Shephard, G. C.. Space-filling zonotopes. Mathematika, 21 (1974). 261–269.CrossRefGoogle Scholar
[Ve54]Venkov, B. A.. On a class of Euclidean poly topes (in Russian). Vestnik Leningradskogo Univ., Ser. math. phys. chem., 9 (1954), 11–31.Google Scholar
[Vo908]Voronoĭ, G. F.. Nouvelles applications des paramètres continus à la théorie de formes quadratiques - Deuxième mémoire. J. reine angew. Math., 134 (1908), 198–287; 136 (1909), 67–178.CrossRefGoogle Scholar
[Wh87]White, N. (ed.), Combinatorial Geometries (Encyclopedia of Mathematics and its Applications 29). Cambridge Univ. Press (1987).Google Scholar
[Zh29]Zhitomirskiĭ, O. K., Verschärfung eines Stazes von Voronoĭ. Zhurnal Leningradskogo Math. Obshtchesiva, 2 (1929), 131–151.Google Scholar