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On the zone theorem for hyperplane arrangements

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(n d−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

Research of Herbert Edelsbrunner was supported by the National Science Foundation under grant CCR-89-21421. Raimund Seidel acknowledges support by an NSF Presidential Young Investigator Grant CCR-90-58440. Micha Sharir has been supported by ONR Grant N00014-90-J-1284, by NSF Grant CCR-89-01484, and by grants from the U.S.-Israeli Binational Science Foundation, the German-Israeli Foundation for Scientific Research and Development, and the Fund for Basic Research of the Israeli Academy of Sciences.

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© 1991 Springer-Verlag Berlin Heidelberg

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Edelsbrunner, H., Seidel, R., Sharir, M. (1991). On the zone theorem for hyperplane arrangements. In: Maurer, H. (eds) New Results and New Trends in Computer Science. Lecture Notes in Computer Science, vol 555. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0038185

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  • DOI: https://doi.org/10.1007/BFb0038185

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-54869-0

  • Online ISBN: 978-3-540-46457-0

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