On the zone theorem for hyperplane arrangements

Herbert Edelsbrunner, Raimund Seidel, Micha Sharir

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

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.

Original languageEnglish
Title of host publicationNew Results and New Trends in Computer Science, Proceedings
EditorsHermann Maurer
PublisherSpringer Verlag
Pages108-123
Number of pages16
ISBN (Print)9783540548690
DOIs
StatePublished - 1991
EventSymposium on New Results and New Trends in Computer Science, 1991 - Graz, Austria
Duration: 20 Jun 199121 Jun 1991

Publication series

NameLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume555 LNCS
ISSN (Print)0302-9743
ISSN (Electronic)1611-3349

Conference

ConferenceSymposium on New Results and New Trends in Computer Science, 1991
Country/TerritoryAustria
CityGraz
Period20/06/9121/06/91

Keywords

  • Arrangements
  • Counting faces
  • Discrete and computational geometry
  • Hyperplanes
  • Induction
  • Sweep
  • Zones

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