Advertisement

Discrete & Computational Geometry

, Volume 1, Issue 2, pp 183–193 | Cite as

Some basic properties of packing and covering constants

  • H. Groemer
Article

Abstract

This article concerns packings and coverings that are formed by the application of rigid motions to the members of a given collectionK of convex bodies. There are two possibilities to construct such packings and coverings: One may permit that the convex bodies fromK are used repeatedly, or one may require that these bodies should be used at most once. In each case one can define the packing and covering constants ofK as, respectively, the least upper bound and the greatest lower bound of the densities of all such packings and coverings. Three theorems are proved. First it is shown that there exist always packings and coverings whose densities are equal to the corresponding packing and covering constants. Then, a quantitative continuity theorem is proved which shows in particular that the packing and covering constants depend, in a certain sense, continuously onK. Finally, a kind of a transference theorem is proved, which enables one to evaluate the packing and covering constants when no repetitions are allowed from the case when repetitions are permitted. Furthermore, various consequences of these theorems are discussed.

Keywords

Convex Body Existence Theorem Discrete Comput Geom Paci Rigid Motion 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. 1.
    L. Danzer, Zerlegbarkeit endlichdimensionaler Räume in kongruente Simplices. Math.-Phys. Semesterber. 15 (1968), 87.MathSciNetzbMATHGoogle Scholar
  2. 2.
    G. Fejes Tóth, New results in the theory of packing and covering. In Convexity and Its Applications, 318–359, Birkhäuser Verlag, Basel, 1983.CrossRefGoogle Scholar
  3. 3.
    L. Fejes Tóth, Lagerungen in der Ebene, auf der Kugel und im Raum. 2. Aufl., Springer-Verlag, Berlin-Heidelberg-New York, 1972.CrossRefzbMATHGoogle Scholar
  4. 4.
    H. Groemer, Über Zerlegungen des Euklidischen Raumes. Math. Zeitschr. 79 (1962), 364–375.MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    H. Groemer, Existenzsätze für Lagerungen im Euklidischen Raum. Math. Zeitschr. 81 (1963), 260–278.MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    H. Groemer, On a covering property of convex sets. Proc. Amer. Math. Soc. 59 (1976), 346–352.MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    H. Groemer, Covering and packing properties of bounded sequences of convex sets, Mathematika 29 (1982), 18–31.MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    H. Groemer, Coverings and packings by sequences of convex sets. In Discrete Geometry and Convexity, 262–278, New York Academy of Sciences, New York, 1985.Google Scholar
  9. 9.
    C. A. Rogers, Packing and Covering. Cambridge University Press, Cambridge, 1964.zbMATHGoogle Scholar

Copyright information

© Springer-Verlag New York Inc. 1986

Authors and Affiliations

  • H. Groemer
    • 1
  1. 1.Department of MathematicsThe University of ArizonaTucson

Personalised recommendations