On the union of κ-round objects in three and four dimensions

Boris Aronov, Alon Efrat, Vladlen Koltun, Micha Sharir

    Research output: Contribution to journalArticle

    Abstract

    A compact set c in ℝd is κ-round if for every point p ∈ ∂ C there exists a closed ball that contains p, is contained in c, and has radius κ diam c. We show that, for any fixed κ > 0, the combinatorial complexity of the union of n κ-round, not necessarily convex, objects in ℝ3 (resp., in ℝ4) of constant description complexity is O(n2+ε) (resp., O(n 3+ε)) for any ε > 0, where the constant of proportionality depends on ε, κ, and the algebraic complexity of the objects. The bound is almost tight in the worst case.

    Original languageEnglish (US)
    Pages (from-to)511-526
    Number of pages16
    JournalDiscrete and Computational Geometry
    Volume36
    Issue number4
    DOIs
    StatePublished - 2006

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    Union
    Algebraic Complexity
    Combinatorial Complexity
    Compact Set
    Ball
    Radius
    Closed
    Object

    ASJC Scopus subject areas

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

    Cite this

    On the union of κ-round objects in three and four dimensions. / Aronov, Boris; Efrat, Alon; Koltun, Vladlen; Sharir, Micha.

    In: Discrete and Computational Geometry, Vol. 36, No. 4, 2006, p. 511-526.

    Research output: Contribution to journalArticle

    Aronov, Boris ; Efrat, Alon ; Koltun, Vladlen ; Sharir, Micha. / On the union of κ-round objects in three and four dimensions. In: Discrete and Computational Geometry. 2006 ; Vol. 36, No. 4. pp. 511-526.
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