Geometric permutations induced by line transversals through a fixed point

Boris Aronov, Shakhar Smorodinsky

    Research output: Contribution to journalArticle

    Abstract

    A line transversal of a family S of n pairwise disjoint convex objects is a. straight line meeting all members of S. A geometric permutation of S is the pair of orders in which members of S are met by a line transversal, one order being the reverse of the other. In this note we consider a long-standing open problem in transversal theory, namely, that of determining the largest number of geometric permutations that a family of n pairwise disjoint convex objects in ℝd can admit. We settle a restricted variant of this problem. Specifically, we show that the maximum number of those geometric permutations to a family of n > 2 pairwise-disjoint convex objects that are induced by lines passing through any fixed point is between K(n ?1, d ?1) and K(n, d ?1), where K(n, d) Σi=0 d (i n?1) = Θ(nd) is the number of pairs of antipodal cells in a simple arrangement of n great (d ?1)-spheres in a d-sphere. By a similar argument, we show that the maximum number of connected components of the space of all line transversals through a fixed point to a family of n > 2 possibly intersecting convex objects is K(n, d ? 1). Finally, we refute a conjecture of Sharir and Smorodinsky on the number of neighbor pairs in geometric permutations and offer an alternative conjecture which may be a first step towards solving the aforementioned general problem of bounding the number of geometric permutations.

    Original languageEnglish (US)
    Pages (from-to)285-294
    Number of pages10
    JournalDiscrete and Computational Geometry
    Volume34
    Issue number2
    DOIs
    StatePublished - 2005

    Fingerprint

    Transversals
    Permutation
    Fixed point
    Line
    Pairwise
    Disjoint
    Connected Components
    Straight Line
    Reverse
    Open Problems
    Arrangement
    Family
    Object
    Alternatives
    Cell

    ASJC Scopus subject areas

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

    Cite this

    Geometric permutations induced by line transversals through a fixed point. / Aronov, Boris; Smorodinsky, Shakhar.

    In: Discrete and Computational Geometry, Vol. 34, No. 2, 2005, p. 285-294.

    Research output: Contribution to journalArticle

    Aronov, Boris ; Smorodinsky, Shakhar. / Geometric permutations induced by line transversals through a fixed point. In: Discrete and Computational Geometry. 2005 ; Vol. 34, No. 2. pp. 285-294.
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