Skip to main content
SpringerLink
Log in

The Voronoi Conjecture for Parallelohedra with Simply Connected \(\delta \)-Surfaces

  • Published:
Discrete & Computational Geometry Aims and scope Submit manuscript

Abstract

We show that the Voronoi conjecture is true for parallelohedra with simply connected \(\delta \)-surfaces. That is, we show that if the boundary of parallelohedron \(P\) remains simply connected after removing closed nonprimitive faces of codimension 2, then \(P\) is affinely equivalent to a Dirichlet–Voronoi domain of some lattice. Also, we construct the \(\pi \)-surface associated with a parallelohedron and give another condition in terms of a homology group of the constructed surface. Every parallelohedron with a simply connected \(\delta \)-surface also satisfies the condition on the homology group of the \(\pi \)-surface.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4

Similar content being viewed by others

References

  1. Delone, B.: Sur la partition régulière de l’espace à 4 dimensions. Bull. Acad. Sci. URSS. VII Sér. Première Partie 1, 79–110 (1929), Deuxième Partie 2, 147–164 (1929)

  2. Delone, B.: Sur la sphère vide. A la mémoire de Georges Voronoi. Bull. Acad. Sci. URSS. Cl. Sci. Math. 6, 793–800 (1934)

  3. Deza, M., Grishukhin, V.: Properties of parallelotopes equivalent to Voronoi’s conjecture. Eur. J. Comb. 25, 517–533 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  4. Dienst, T.: Fedorov’s five parallelohedra in \(\mathbb{R}^3\). http://www.matha.mathematik.uni-dortmund.de/thilo/contents/fedorov.htm

  5. Erdahl, R.: Zonotopes, dicings, and Voronoi’s conjecture on parallelohedra. Eur. J. Comb. 20(6), 527–549 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  6. Garber, A.: On \(\pi \)-surfaces of four-dimensional parallelohedra. http://arxiv.org/abs/1309.7661 (2013)

  7. Gavrilyuk, A.: Geometry of lifts of tilings of Euclidean spaces (preprint). http://arxiv.org/abs/1501.06088 (2015)

  8. Gavrilyuk, A.A.: A class of affinely equivalent Voronoi parallelohedra. Math. Notes 95(5), 625–633 (2014)

    Article  MathSciNet  Google Scholar 

  9. Gruber, P.M., Lekkerkerker, C.G.: Geometry of Numbers, 2nd edn. North-Holland, Amsterdam (1987)

    MATH  Google Scholar 

  10. Hatcher, A.: Algebraic Topology. Cambridge University Press, Cambridge (2002)

    MATH  Google Scholar 

  11. Ordine, A., Magazinov, A.: A criterion of reducibility for a parallelohedron (in Russian). http://arxiv.org/abs/1305.5345 (2013)

  12. McMullem, P.: On zonotopes. Trans. Amer. Math. Soc. 159, 91–109 (1971)

    Article  MathSciNet  Google Scholar 

  13. McMullen, P.: Convex bodies which tile space by translation. Mathematika 27(1), 113–121 (1980)

    Article  MATH  MathSciNet  Google Scholar 

  14. McMullen, P.: Convex bodies which tile space by translation: acknowledgement of priority. Mathematika 28(2), 191–191 (1981)

    Article  MATH  MathSciNet  Google Scholar 

  15. Minkowski, H.: Allgemeine Lehrsätze über die convexen Polyeder. Gött. Nachr. 198–219 (1897)

  16. Ordine, A.: Proof of the Voronoi conjecture on parallelotopes in a new special case. Ph.D. thesis, Queen’s University, Ontario. http://higeom.math.msu.su/people/garber/Ordine.pdf8 (2005)

  17. Ryshkov, S., Rybnikov Jr, K.: The theory of quality translations with applications to tilings. Eur. J. Comb. 18(4), 431–444 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  18. Venkov, B.A.: About one class of Euclidean polytopes (in Russian). Vestnik Leningr. Univ. Ser. Math. Phys. Chem. 9, 11–31 (1954)

    MathSciNet  Google Scholar 

  19. Voronoi, G.: Nouvelles applications des paramètres continus à la théorie des formes quadratiques. Crelle’s J. 133, 97–178 (1908), 134, 198–287 (1908) and 136, 67–181 (1909)

  20. Zhitomirskii, O.K.: Verschärfung eines Satzes von Woronoi. J. Leningr. Math. Soc. 2, 131–151 (1929)

    Google Scholar 

Download references

Acknowledgments

The authors would like to thank Professor Nikolai Dolbilin and Professor Robert Erdahl for fruitful discussions and helpful comments and remarks. The first author is supported by RScF project 14-11-00414.

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Texas at Brownsville, One University Blvd., Brownsville, TX, 78520, USA

    A. Garber

  2. Higher School of Economics, Faculty of Computer Science, 3 Kochnovskiy Proezd, Moscow, 125319, Russia

    A. Gavrilyuk

  3. Center for Optical Neural Technologies, Scientific Research Institute for Systems Analysis, RAS, Vavilova Str., 44, Korp. 2, Moscow, 119333, Russia

    A. Gavrilyuk

  4. Steklov Mathematical Institute, 8 Gubkina Str., Moscow, 119991, Russia

    A. Magazinov

Authors
  1. A. Garber
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. A. Gavrilyuk
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. A. Magazinov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to A. Garber.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Garber, A., Gavrilyuk, A. & Magazinov, A. The Voronoi Conjecture for Parallelohedra with Simply Connected \(\delta \)-Surfaces. Discrete Comput Geom 53, 245–260 (2015). https://doi.org/10.1007/s00454-014-9660-z

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00454-014-9660-z

Keywords

Search

Navigation