On geometric permutations induced by lines transversal through a fixed point

Boris Aronov, Shakhar Smorodinsky

    Research output: Contribution to conferencePaperpeer-review

    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 - l, d - 1) and K(n, d - 1), where K(n, d) = Σid=0 (n-1 i) = ⊙(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 lines transversal 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)
    Pages251-256
    Number of pages6
    StatePublished - 2005
    EventSixteenth Annual ACM-SIAM Symposium on Discrete Algorithms - Vancouver, BC, United States
    Duration: Jan 23 2005Jan 25 2005

    Other

    OtherSixteenth Annual ACM-SIAM Symposium on Discrete Algorithms
    Country/TerritoryUnited States
    CityVancouver, BC
    Period1/23/051/25/05

    ASJC Scopus subject areas

    • Software
    • Mathematics(all)

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