On Center Regions and Balls Containing Many Points

  • Shakhar Smorodinsky
  • Marek Sulovský
  • Uli Wagner
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5092)


We study the disk containment problem introduced by Neumann-Lara and Urrutia and its generalization to higher dimensions. We relate the problem to centerpoints and lower centerpoints of point sets. Moreover, we show that for any set of n points in Open image in new window , there is a subset A ⊆ S of size \(\lfloor \frac{d+3}{2}\rfloor\) such that any ball containing A contains at least roughly \(\frac{4}{5ed^3}n\) points of S. This improves previous bounds for which the constant was exponentially small in d. We also consider a generalization of the planar disk containment problem to families of pseudodisks.


Convex Body Moment Curve Convex Position Halfspace Depth Containment Problem 
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  1. 1.
    Ábrego, B.M., Fernández-Merchant, S.: A Lower Bound for the Rectilinear Crossing Number. Graphs Comb. 21(3), 293–300 (2005)CrossRefzbMATHGoogle Scholar
  2. 2.
    Agarwal, P.K., Sharir, M., Welzl, E.: Algorithms for Center and Tverberg Points. In: Proceedings of the Twentieth Annual Symposium on Computational Geometry (SoCG), New York, pp. 61–67 (2004)Google Scholar
  3. 3.
    Aicholzer, O., Garcia, J., Orden, D., Ramos, P.: New Lower Bounds for the Number of ( ≤ k)-Edges and the Rectilinear Crossing Number of k n. In: Proceedings of the 18th ACM-SIAM Symposium on Discrete Algorithms (2007)Google Scholar
  4. 4.
    Andrzejak, A., Welzl, E.: In Between k-Sets, j-Facets, and i-Faces: (i,j)-Partitions. Discrete Comput. Geom. 29(1), 105–131 (2003)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Bárány, I., Schmerl, J.H., Sidney, S.J., Urrutia, J.: A Combinatorial Result about Points and Balls in Euclidean Space. Discrete Comput. Geom. 4(3), 259–262 (1989)CrossRefMathSciNetzbMATHGoogle Scholar
  6. 6.
    Clarkson, K.L., Shor, P.W.: Applications of Random Sampling in Computational Geometry. II. Discrete Comput. Geom. 4(5), 387–421 (1989)CrossRefMathSciNetzbMATHGoogle Scholar
  7. 7.
    Edelsbrunner, H., Hasan, N., Seidel, R., Shen, X.J.: Circles Through Two Points that Always Enclose Many Points. Geom. Dedicata 32(1), 1–12 (1989)CrossRefMathSciNetzbMATHGoogle Scholar
  8. 8.
    Hayward, R.: A Note on the Circle Containment Problem. Discrete Comput. Geom. 4(3), 263–264 (1989)CrossRefMathSciNetzbMATHGoogle Scholar
  9. 9.
    Hayward, R., Rappaport, D., Wenger, R.: Some Extremal Results on Circles Containing Points. Discrete Comput. Geom. 4(3), 253–258 (1989)CrossRefMathSciNetzbMATHGoogle Scholar
  10. 10.
    Lovász, L., Vesztergombi, K., Wagner, U., Welzl, E.: Convex Quadrilaterals and k-Sets. In: Towards a theory of geometric graphs, Contemp. Math., vol. 342, pp. 139–148 (2004)Google Scholar
  11. 11.
    Matoušek, J.: Lectures on Discrete Geometry 212 of Graduate Texts in Mathematics. Springer, New York (2002)Google Scholar
  12. 12.
    Neumann-Lara, V., Urrutia, J.: A Combinatorial Result on Points and Circles on the Plane. Discrete Math 69(2), 173–178 (1988)CrossRefMathSciNetzbMATHGoogle Scholar
  13. 13.
    Smorodinsky, S., Sharir, M.: Selecting Points that Are Heavily Covered by Pseudo-Circles, Spheres or Rectangles. Combin. Probab. Comput. 13(3), 389–411 (2004)CrossRefMathSciNetzbMATHGoogle Scholar
  14. 14.
    Wagner, U.: On a Generalization of the Upper Bound Theorem. In: Proceedings of the 47 Annual Symposium on the Foundations of Computer Science (FOCS), pp. 635–645 (2006)Google Scholar
  15. 15.
    Welzl, E.: Entering and Leaving j-Facets. Discrete Comput. Geom. 25(3), 351–364 (2001)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Shakhar Smorodinsky
    • 1
  • Marek Sulovský
    • 2
  • Uli Wagner
    • 2
  1. 1.Department of MathematicsBen-Gurion UniversityBe’er ShevaIsrael
  2. 2.Institute of Theoretical Computer ScienceETH ZurichZurichSwitzerland

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