k-sets and random hulls

Abstract

We re-examine the probabilistic analysis of Clarkson and Shor [5] involvingk-sets of point sets and related structures. By studying more carefully the equations that they derive, we are able to obtain refined analysis of these quantities, which lead to a collection of interesting relationships involvingk-sets, convex hulls of random samples, and generalizations of these constructs.

This is a preview of subscription content, access via your institution.

References

  1. [1]

    N. Alon, andE. Győri: The number of small semispaces of a finite set of points in the plane,J. Combin. Theory Ser. A 41 (1986), 154–157.

    Google Scholar 

  2. [2]

    B. Aronov, B. Chazelle, H. Edelsbrunner, L. Guibas, M. Sharir andR. Wenger: Points and triangles in the plane and halving planes in space,Discrete Comput. Geom. 6 (1991), 435–442.

    Google Scholar 

  3. [3]

    I. Bárány, Z. Füredi, andL. Lovász: On the number of halving planes,Proc. 5th ACM Symp. on Computational Geometry, 1989, 140–144.

  4. [4]

    Brønstad:An introduction to Convex Polytopes, Springer-Verlag, Heidelberg 1983.

    Google Scholar 

  5. [5]

    K. Clarkson, andP. Shor: Applications of random sampling in computational geometry, II,Discrete Comput. Geom. 4 (1989), 387–421.

    Google Scholar 

  6. [6]

    H. Edelsbrunner, andE. Welzl: On the number of line separations of a finite set in the plane,J. Combin. Theory Ser. A 38 (1985), 15–29.

    Google Scholar 

  7. [7]

    P. Erdős, L. Lovász, A. Simmons, andE. G. Strauss: Dissection graphs of planar point sets, InA Survey of Combinatorial Theory, J. N. Srivastava et al., eds., North-Holland, Amsterdam, 1973, 139–149.

    Google Scholar 

  8. [8]

    J. E. Goodman, andR. Pollack: On the number ofk-subsets of a set ofn points in the plane,J. Combin. Theory, Ser. A 36 (1984), 101–104.

    Google Scholar 

  9. [9]

    R. Graham, D. Knuth, andO. Patashnik:Concrete Mathematics, Addison-Wesley, Reading, MA, 1989.

    Google Scholar 

  10. [10]

    L. Guibas, D. Knuth, andM. Sharir: Randomized incremental construction of Delaunay and Voronoi Diagrams,Algorithmica 7 (1992), 381–413.

    Google Scholar 

  11. [11]

    L. Lovász: On the number of halving lines,Ann. Univ. Sci. Budapest, Eötvös, Sect. Math. 14 (1971), 107–108.

    Google Scholar 

  12. [12]

    C. Ó'Dúnlaing, K. Mehlhorn, andS. Meiser: Abstract Voronoi diagrams, manuscript, 1989.

  13. [13]

    J. Pach, W. Steiger, andE. Szemerédi: An upper bound on the number of planark-sets,Proc. 30th IEEE Symp. on Foundations of Computer Science, 1989, 72–79.

  14. [14]

    M. Sharir: Onk-sets in arrangements of curves and surfaces,Discrete Comput. Geom. 6 (1991), 593–613.

    Google Scholar 

  15. [15]

    E. Yaniv: Randomized incremental construction of Delaunay triangulations: Theory and practice, M. Sc. thesis, Tel Aviv University, Tel Aviv, Israel, 1991.

    Google Scholar 

Download references

Author information

Affiliations

Authors

Additional information

Work on this paper has been supported by Office of Naval Research Grant N00014-89-J-3042 and N00014-90-J-1284, by National Science Foundation Grant CCR-89-01484, and by grants from the U.S.-Israeli Binational Science Foundation, the Fund for Basic Research administered by the Israeli Academy of Sciences, and the G.I.F., the German-Israeli Foundation for Scientific Research and Development.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Sharir, M. k-sets and random hulls. Combinatorica 13, 483–495 (1993). https://doi.org/10.1007/BF01303520

Download citation

AMS subject classification code (1991)

  • 52 A 22
  • 52 A 37
  • 05 C 99