Discrete & Computational Geometry

, Volume 5, Issue 4, pp 389–397

Double-lattice packings of convex bodies in the plane

  • G. Kuperberg
  • W. Kuperberg
Article

Abstract

Mahler [7] and Fejes Tóth [2] proved that every centrally symmetric convex plane bodyK admits a packing in the plane by congruent copies ofK with density at least √3/2. In this paper we extend this result to all, not necessarily symmetric, convex plane bodies. The methods of Mahler and Fejes Tóth are constructive and produce lattice packings consisting of translates ofK. Our method is constructive as well, and it produces double-lattice packings consisting of translates ofK and translates of−K. The lower bound of √3/2 for packing densities produced here is an improvement of the bounds obtained previously in [5] and [6].

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    G. Blind, Problem No. 34, Research problems,Period. Math. Hungar. 14 (1983), 309–312.MathSciNetCrossRefGoogle Scholar
  2. 2.
    L. Fejes Tóth, On the densest packing of domains,Proc. Kon. Ned. Akad. Wet. 51 (1948), 189–192.Google Scholar
  3. 3.
    L. Fejes Tóth, Some packing and covering theorems,Acta Sci. Math. (Szeged) 12/A (1950), 62–67.MathSciNetMATHGoogle Scholar
  4. 4.
    L. Fejes Tóth,Lagerungen in der Ebene, auf der Kugel und im Raum, Springer-Verlag, Berlin, Göttingen, Heidelberg, 1953.CrossRefMATHGoogle Scholar
  5. 5.
    W. Kuperberg, Packing convex bodies in the plane with density greater than 3/4,Geom. Dedicata 13 (1982), 149–155.MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    W. Kuperberg, On packing the plane with congruent copies of a convex body,Colloq. Math. Soc. János Bolyai 48, Intuitive Geometry, Siófok, 1985, pp. 317–329.Google Scholar
  7. 7.
    K. Mahler, The theorem of Minkowski-Hlawka,Duke Math. J. 13 (1946), 611–621.MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    K. Reinhardt, Über die dichteste gitterförmige Lagerung kongruente Bereiche in der Ebene und eine besondere Art convexe Kurven,Abh. Math. Sem. Univ. Hamburg 10 (1934), 216–230.MathSciNetCrossRefGoogle Scholar
  9. 9.
    C. A. Rogers, The closest packing of convex two-dimensional domains,Acta Math. 86 (1951), 309–321.MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    C. A. Rogers,Packing and Covering, Cambridge Tracts 54, Cambridge University Press, Cambridge, 1964.MATHGoogle Scholar

Copyright information

© Springer-Verlag New York Inc. 1990

Authors and Affiliations

  • G. Kuperberg
    • 1
  • W. Kuperberg
    • 2
  1. 1.Department of MathematicsUniversity of California at BerkeleyBerkeleyUSA
  2. 2.Division of MathematicsAuburn UniversityAuburnUSA

Personalised recommendations