On the sum of squares of cell complexities in hyperplane arrangements

Boris Aronov, Jiří Matoušek, Micha Sharir

    Research output: Contribution to journalArticle

    Abstract

    Let H be a collection of n hyperplanes in Rd, d≥2. For each cell c of the arrangement of H let fi(c) denote the number of faces of c of dimension i, and let f(c) = ∑i=0 d-1 fi(c). We prove that ∑c f(c)2 = O(ndlog d 2-1 n), where the sum extends over all cells of the arrangement. Among other applications, we show that the total number of faces bounding any m distinct cells in an arrangement of n hyperplanes in Rd is O(m 1 2n d 2log ( d 2-1) 2 n) and provide a lower bound on the maximum possible face count in m distinct cells, which is close to the upper bound, and for many values of m and n is Ω(m 1 2n d 2).

    Original languageEnglish (US)
    Pages (from-to)311-321
    Number of pages11
    JournalJournal of Combinatorial Theory, Series A
    Volume65
    Issue number2
    DOIs
    StatePublished - 1994

    Fingerprint

    Hyperplane Arrangement
    Sum of squares
    Arrangement
    Cell
    Face
    Hyperplane
    Distinct
    Count
    Lower bound
    Upper bound
    Denote

    ASJC Scopus subject areas

    • Discrete Mathematics and Combinatorics
    • Theoretical Computer Science

    Cite this

    On the sum of squares of cell complexities in hyperplane arrangements. / Aronov, Boris; Matoušek, Jiří; Sharir, Micha.

    In: Journal of Combinatorial Theory, Series A, Vol. 65, No. 2, 1994, p. 311-321.

    Research output: Contribution to journalArticle

    Aronov, Boris ; Matoušek, Jiří ; Sharir, Micha. / On the sum of squares of cell complexities in hyperplane arrangements. In: Journal of Combinatorial Theory, Series A. 1994 ; Vol. 65, No. 2. pp. 311-321.
    @article{227903090659498f858b2e59e87f26b2,
    title = "On the sum of squares of cell complexities in hyperplane arrangements",
    abstract = "Let H be a collection of n hyperplanes in Rd, d≥2. For each cell c of the arrangement of H let fi(c) denote the number of faces of c of dimension i, and let f(c) = ∑i=0 d-1 fi(c). We prove that ∑c f(c)2 = O(ndlog d 2-1 n), where the sum extends over all cells of the arrangement. Among other applications, we show that the total number of faces bounding any m distinct cells in an arrangement of n hyperplanes in Rd is O(m 1 2n d 2log ( d 2-1) 2 n) and provide a lower bound on the maximum possible face count in m distinct cells, which is close to the upper bound, and for many values of m and n is Ω(m 1 2n d 2).",
    author = "Boris Aronov and Jiř{\'i} Matoušek and Micha Sharir",
    year = "1994",
    doi = "10.1016/0097-3165(94)90027-2",
    language = "English (US)",
    volume = "65",
    pages = "311--321",
    journal = "Journal of Combinatorial Theory - Series A",
    issn = "0097-3165",
    publisher = "Academic Press Inc.",
    number = "2",

    }

    TY - JOUR

    T1 - On the sum of squares of cell complexities in hyperplane arrangements

    AU - Aronov, Boris

    AU - Matoušek, Jiří

    AU - Sharir, Micha

    PY - 1994

    Y1 - 1994

    N2 - Let H be a collection of n hyperplanes in Rd, d≥2. For each cell c of the arrangement of H let fi(c) denote the number of faces of c of dimension i, and let f(c) = ∑i=0 d-1 fi(c). We prove that ∑c f(c)2 = O(ndlog d 2-1 n), where the sum extends over all cells of the arrangement. Among other applications, we show that the total number of faces bounding any m distinct cells in an arrangement of n hyperplanes in Rd is O(m 1 2n d 2log ( d 2-1) 2 n) and provide a lower bound on the maximum possible face count in m distinct cells, which is close to the upper bound, and for many values of m and n is Ω(m 1 2n d 2).

    AB - Let H be a collection of n hyperplanes in Rd, d≥2. For each cell c of the arrangement of H let fi(c) denote the number of faces of c of dimension i, and let f(c) = ∑i=0 d-1 fi(c). We prove that ∑c f(c)2 = O(ndlog d 2-1 n), where the sum extends over all cells of the arrangement. Among other applications, we show that the total number of faces bounding any m distinct cells in an arrangement of n hyperplanes in Rd is O(m 1 2n d 2log ( d 2-1) 2 n) and provide a lower bound on the maximum possible face count in m distinct cells, which is close to the upper bound, and for many values of m and n is Ω(m 1 2n d 2).

    UR - http://www.scopus.com/inward/record.url?scp=38149147699&partnerID=8YFLogxK

    UR - http://www.scopus.com/inward/citedby.url?scp=38149147699&partnerID=8YFLogxK

    U2 - 10.1016/0097-3165(94)90027-2

    DO - 10.1016/0097-3165(94)90027-2

    M3 - Article

    AN - SCOPUS:38149147699

    VL - 65

    SP - 311

    EP - 321

    JO - Journal of Combinatorial Theory - Series A

    JF - Journal of Combinatorial Theory - Series A

    SN - 0097-3165

    IS - 2

    ER -