Hostname: page-component-745bb68f8f-f46jp Total loading time: 0 Render date: 2025-02-04T14:37:46.620Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Get access

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

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