Piercing Translates and Homothets of a Convex Body

  • Adrian Dumitrescu
  • Minghui Jiang
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5757)


According to a classical result of Grünbaum, the transversal number \(\tau({\mathcal F})\) of any family \({\mathcal F}\) of pairwise-intersecting translates or homothets of a convex body C in ℝ d is bounded by a function of d. Denote by α(C) (resp. β(C)) the supremum of the ratio of the transversal number \(\tau({\mathcal F})\) to the packing number \(\nu({\mathcal F})\) over all families \({\mathcal F}\) of translates (resp. homothets) of a convex body C in ℝ d . Kim et al. recently showed that α(C) is bounded by a function of d for any convex body C in ℝ d , and gave the first bounds on α(C) for convex bodies C in ℝ d and on β(C) for convex bodies C in the plane. In this paper, we show that β(C) is also bounded by a function of d for any convex body C in ℝ d , and present new or improved bounds on both α(C) and β(C) for various convex bodies C in ℝ d for all dimensions d. Our techniques explore interesting inequalities linking the covering and packing densities of a convex body. Our methods for obtaining upper bounds are constructive and lead to efficient constant-factor approximation algorithms for finding a minimum-cardinality point set that pierces a set of translates or homothets of a convex body.


Lattice Point Convex Body Lattice Packing Symmetric Convex Symmetric Convex Body 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Alon, N., Kleitman, D.J.: Piercing convex sets and the Hadwiger-Debrunner (p,q)-problem. Advances in Mathematics 96, 103–112 (1992)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Bereg, S., Dumitrescu, A., Jiang, M.: On covering problems of Rado. Algorithmica, doi:10.1007/s00453-009-9298-z (to appear); A preliminary version in: Proceedings of the 11th Scandinavian Workshop on Algorithm Theory, pp. 294–305 (2008)Google Scholar
  3. 3.
    Braß, P., Moser, W., Pach, J.: Research Problems in Discrete Geometry. Springer, New York (2005)MATHGoogle Scholar
  4. 4.
    Chakerian, G.D., Stein, S.K.: Some intersection properties of convex bodies. Proceedings of the American Mathematical Society 18, 109–112 (1967)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Chan, T.: Polynomial-time approximation schemes for packing and piercing fat objects. Journal of Algorithms 46, 178–189 (2003)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Danzer, L.: Zur Lösung des Gallaischen Problems über Kreisscheiben in der Euklidischen Ebene. Studia Scientiarum Mathematicarum Hungarica 21, 111–134 (1986)MathSciNetMATHGoogle Scholar
  7. 7.
    Danzer, L., Grünbaum, B., Klee, V.: Helly’s theorem and its relatives. In: Proceedings of Symposia in Pure Mathematics., vol. 7, pp. 101–181. American Mathematical Society (1963)Google Scholar
  8. 8.
    Fon-Der-Flaass, D.G., Kostochka, A.V.: Covering boxes by points. Discrete Mathematics 120, 269–275 (1993)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Fowler, R.J., Paterson, M.S., Tanimoto, S.L.: Optimal packing and covering in the plane are NP-complete. Information Processing Letters 12, 133–137 (1981)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Grünbaum, B.: On intersections of similar sets. Portugaliae Mathematica 18, 155–164 (1959)MathSciNetMATHGoogle Scholar
  11. 11.
    Gyárfás, A., Lehel, J.: Covering and coloring problems for relatives of intervals. Discrete Mathematics 55, 167–180 (1985)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Karasev, R.N.: Transversals for families of translates of a two-dimensional convex compact set. Discrete and Computational Geometry 24, 345–353 (2000)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Károlyi, G.: On point covers of parallel rectangles. Periodica Mathematica Hungarica 23, 105–107 (1991)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Kim, S.-J., Nakprasit, K., Pelsmajer, M.J., Skokan, J.: Transversal numbers of translates of a convex body. Discrete Mathematics 306, 2166–2173 (2006)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Rogers, C.A.: A note on coverings. Mathematika 4, 1–6 (1957)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Schmidt, W.M.: On the Minkowski-Hlawka theorem. Illinois Journal of Mathematics 7, 18–23 (1963)MathSciNetGoogle Scholar
  17. 17.
    Wenger, R.: Helly-type theorems and geometric transversals. In: Handbook of Discrete and Computational Geometry, 2nd edn., pp. 73–96. CRC Press, Boca Raton (2004)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • Adrian Dumitrescu
    • 1
  • Minghui Jiang
    • 2
  1. 1.Department of Computer ScienceUniversity of Wisconsin-MilwaukeeUSA
  2. 2.Department of Computer ScienceUtah State UniversityLoganUSA

Personalised recommendations