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)


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.


Discrete and computational geometry arrangements hyperplanes zones counting faces induction sweep 


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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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