Polytopes in arrangements

Boris Aronov, Tamal K. Dey

    Research output: Chapter in Book/Report/Conference proceedingChapter

    Abstract

    Consider an arrangement of n hyperplanes in Rd. Families of convex polytopes whose boundaries are contained in the union of the hyperplanes are the subject of this paper. We aim to bound their combinatorial complexity. Exact asymptotic bounds were known for the case where the polytopes are cells of the arrangement. Situations where the polytopes are pairwise openly disjoint have also been considered in the past. However, no non-trivial bound was known for the general case where the polytopes may have overlapping interiors. We analyze families of polytopes that do not share vertices. In R3 we show an O(k1/3n) bound on the number of faces of k such polytopes. We also discuss worst-case lower bounds and higher-dimensional versions of the problem. Among other results, we show that the maximum number of facets of k pairwise vertex-disjoint polytopes in Rd is Ω(k1/2nd/2) which is a factor of √n away from the best known upper bound in the range nd-2≤k≤nd.

    Original languageEnglish (US)
    Title of host publicationProceedings of the Annual Symposium on Computational Geometry
    PublisherACM
    Pages154-162
    Number of pages9
    StatePublished - 1999
    EventProceedings of the 1999 15th Annual Symposium on Computational Geometry - Miami Beach, FL, USA
    Duration: Jun 13 1999Jun 16 1999

    Other

    OtherProceedings of the 1999 15th Annual Symposium on Computational Geometry
    CityMiami Beach, FL, USA
    Period6/13/996/16/99

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    ASJC Scopus subject areas

    • Chemical Health and Safety
    • Software
    • Safety, Risk, Reliability and Quality
    • Geometry and Topology

    Cite this

    Aronov, B., & Dey, T. K. (1999). Polytopes in arrangements. In Proceedings of the Annual Symposium on Computational Geometry (pp. 154-162). ACM.