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On the zone theorem for hyperplane arrangements

  • Herbert Edelsbrunner
  • Raimund Seidel
  • Micha Sharir
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 555)

Abstract

The zone theorem for an arrangement of n hyperplanes in d-dimensional real space says that the total number of faces bounding the cells intersected by another hyperplane is O(nd−1). This result is the basis of a time-optimal incremental algorithm that constructs a hyperplane arrangement and has a host of other algorithmic and combinatorial applications. Unfortunately, the original proof of the zone theorem, for d ≥ 3, turned out to contain a serious and irreparable error. This paper presents a new proof of the theorem. Our proof is based on an inductive argument, which also applies in the case of pseudo-hyperplane arrangements. We also briefly discuss the fallacies of the old proof along with some ways of partially saving that approach.

Keywords

Discrete and computational geometry arrangements hyperplanes zones counting faces induction sweep 

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References

  1. [1]
    B. Aronov, J. Matoušek and M. Sharir. On the sum of squares of cell complexities in hyperplane arrangements. In preparation.Google Scholar
  2. [2]
    M. Bern, D. Eppstein, P. Plassmann and F. Yao. Horizon theorems for lines and polygons. Manuscript, XEROX Palo Alto Research Center, California, 1989.Google Scholar
  3. [3]
    A. Björner, M. Las Vergnas, B. Sturmfels, N. White and G. Ziegler. Oriented Matroids. Book in preparation.Google Scholar
  4. [4]
    B. Chazelle, L. J. Guibas and D. T. Lee. The power of geometric duality. BIT 25 (1985), 76–90.Google Scholar
  5. [5]
    H. Edelsbrunner. Algorithms in Combinatorial Geometry. Springer-Verlag, Heidelberg, Germany, 1987.Google Scholar
  6. [6]
    H. Edelsbrunner. The upper envelope of piecewise linear functions: tight bounds on the number of faces. Discrete Comput. Geom. 4 (1989), 337–343.Google Scholar
  7. [7]
    H. Edelsbrunner and L. J. Guibas. Topologically sweeping an arrangement. J. Comput System Sci. 38 (1989), 165–194.Google Scholar
  8. [8]
    H. Edelsbrunner, L. J. Guibas, J. Pach, R. Pollack, R. Seidel and M. Sharir. Arrangements of curves in the plane: topology, combinatorics and algorithms. Proc. 15th Int. Colloq. on Automata, Languages and Programming, 1988, pp. 214–229.Google Scholar
  9. [9]
    H. Edelsbrunner, L. J. Guibas and M. Sharir. The complexity and construction of many faces in an arrangement of lines or of segments. Discrete Comput. Geom. 5 (1990), 161–196.Google Scholar
  10. [10]
    H. Edelsbrunner, J. O'Rourke and R. Seidel. Constructing arrangements of lines and hyperplanes with applications. SIAM J. Comput. 15 (1986), 341–363.Google Scholar
  11. [11]
    B. Grünbaum. Convex Polytopes. John Wiley &. Sons, London, 1967.Google Scholar
  12. [12]
    M. Houle. A note on hyperplane arrangements. Manuscript, University of Tokyo, 1987.Google Scholar
  13. [13]
    J. Matoušek. A simple proof of the weak zone theorem. Manuscript, Charles University, Prague, 1990.Google Scholar
  14. [14]
    J. Pach and M. Sharir. The upper envelope of piecewise linear functions and the boundary of a region enclosed by convex plates: combinatorial analysis. Discrete Comput. Geom. 4 (1989), 291–309.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1991

Authors and Affiliations

  • Herbert Edelsbrunner
    • 1
  • Raimund Seidel
    • 2
  • Micha Sharir
    • 3
    • 4
  1. 1.Department of Computer ScienceUniversity of Illinois at Urbana-ChampaignUrbanaUSA
  2. 2.Department of Electrical Engineering and Computer ScienceUniversity of CaliforniaBerkeleyUSA
  3. 3.School of Mathematical SciencesTel Aviv UniversityTel AvivIsrael
  4. 4.Courant Institute of Mathematical SciencesNew York UniversityNew YorkUSA

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