Unions of fat convex polytopes have short skeletons

Boris Aronov, Mark de Berg

    Research output: Contribution to journalArticle

    Abstract

    The skeleton of a polyhedral set is the union of its edges and vertices. Let P be a set of fat, convex polytopes in three dimensions with n vertices in total, and let f max be the maximum complexity of any face of a polytope in P. We prove that the total length of the skeleton of the union of the polytopes in P is at most O(α(n)· log * n · log f max) times the sum of the skeleton lengths of the individual polytopes.

    Original languageEnglish (US)
    Pages (from-to)53-64
    Number of pages12
    JournalDiscrete and Computational Geometry
    Volume48
    Issue number1
    DOIs
    StatePublished - Jul 2012

    Fingerprint

    Convex Polytopes
    Oils and fats
    Skeleton
    Union
    Polytopes
    Polyhedral Sets
    Polytope
    Three-dimension
    Face

    Keywords

    • Combinatorial complexity
    • Convex polytopes
    • Fat polytopes
    • Skeleton of union

    ASJC Scopus subject areas

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

    Cite this

    Unions of fat convex polytopes have short skeletons. / Aronov, Boris; de Berg, Mark.

    In: Discrete and Computational Geometry, Vol. 48, No. 1, 07.2012, p. 53-64.

    Research output: Contribution to journalArticle

    Aronov, Boris ; de Berg, Mark. / Unions of fat convex polytopes have short skeletons. In: Discrete and Computational Geometry. 2012 ; Vol. 48, No. 1. pp. 53-64.
    @article{ba126b6f0a104a87a23249403dd0b40c,
    title = "Unions of fat convex polytopes have short skeletons",
    abstract = "The skeleton of a polyhedral set is the union of its edges and vertices. Let P be a set of fat, convex polytopes in three dimensions with n vertices in total, and let f max be the maximum complexity of any face of a polytope in P. We prove that the total length of the skeleton of the union of the polytopes in P is at most O(α(n)· log * n · log f max) times the sum of the skeleton lengths of the individual polytopes.",
    keywords = "Combinatorial complexity, Convex polytopes, Fat polytopes, Skeleton of union",
    author = "Boris Aronov and {de Berg}, Mark",
    year = "2012",
    month = "7",
    doi = "10.1007/s00454-012-9422-8",
    language = "English (US)",
    volume = "48",
    pages = "53--64",
    journal = "Discrete and Computational Geometry",
    issn = "0179-5376",
    publisher = "Springer New York",
    number = "1",

    }

    TY - JOUR

    T1 - Unions of fat convex polytopes have short skeletons

    AU - Aronov, Boris

    AU - de Berg, Mark

    PY - 2012/7

    Y1 - 2012/7

    N2 - The skeleton of a polyhedral set is the union of its edges and vertices. Let P be a set of fat, convex polytopes in three dimensions with n vertices in total, and let f max be the maximum complexity of any face of a polytope in P. We prove that the total length of the skeleton of the union of the polytopes in P is at most O(α(n)· log * n · log f max) times the sum of the skeleton lengths of the individual polytopes.

    AB - The skeleton of a polyhedral set is the union of its edges and vertices. Let P be a set of fat, convex polytopes in three dimensions with n vertices in total, and let f max be the maximum complexity of any face of a polytope in P. We prove that the total length of the skeleton of the union of the polytopes in P is at most O(α(n)· log * n · log f max) times the sum of the skeleton lengths of the individual polytopes.

    KW - Combinatorial complexity

    KW - Convex polytopes

    KW - Fat polytopes

    KW - Skeleton of union

    UR - http://www.scopus.com/inward/record.url?scp=84860887358&partnerID=8YFLogxK

    UR - http://www.scopus.com/inward/citedby.url?scp=84860887358&partnerID=8YFLogxK

    U2 - 10.1007/s00454-012-9422-8

    DO - 10.1007/s00454-012-9422-8

    M3 - Article

    VL - 48

    SP - 53

    EP - 64

    JO - Discrete and Computational Geometry

    JF - Discrete and Computational Geometry

    SN - 0179-5376

    IS - 1

    ER -