Polytopes in arrangements

Boris Aronov, Tamal K. Dey

    Research output: Chapter in Book/Report/Conference proceedingChapter


    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
    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


    OtherProceedings of the 1999 15th Annual Symposium on Computational Geometry
    CityMiami Beach, FL, USA

    ASJC Scopus subject areas

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


    Dive into the research topics of 'Polytopes in arrangements'. Together they form a unique fingerprint.

    Cite this