Counting facets and incidences

Pankaj K. Agarwal, Boris Aronov

    Research output: Contribution to journalArticle

    Abstract

    We show that m distinct cells in an arrangement of n planes in ℝ3 are bounded by O(m2/3n+n2) faces, which in turn yields a tight bound on the maximum number of facets bounding m cells in an arrangement of n hyperplanes in ℝd, for every d≥3. In addition, the method is extended to obtain tight bounds on the maximum number of faces on the boundary of all nonconvex cells in an arrangement of triangles in ℝ3. We also present a simpler proof of the O(m2/3nd/3+nd-1) bound on the number of incidences between n hyperplanes in ℝd and m vertices of their arrangement.

    Original languageEnglish (US)
    Pages (from-to)359-369
    Number of pages11
    JournalDiscrete and Computational Geometry
    Volume7
    Issue number1
    DOIs
    StatePublished - Dec 1992

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    Facet
    Counting
    Arrangement
    Incidence
    Hyperplane
    Cell
    Face
    Triangle
    Distinct

    ASJC Scopus subject areas

    • Theoretical Computer Science
    • Geometry and Topology
    • Discrete Mathematics and Combinatorics
    • Computational Theory and Mathematics

    Cite this

    Counting facets and incidences. / Agarwal, Pankaj K.; Aronov, Boris.

    In: Discrete and Computational Geometry, Vol. 7, No. 1, 12.1992, p. 359-369.

    Research output: Contribution to journalArticle

    Agarwal, Pankaj K. ; Aronov, Boris. / Counting facets and incidences. In: Discrete and Computational Geometry. 1992 ; Vol. 7, No. 1. pp. 359-369.
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