A Helly-type theorem for higher-dimensional transversals

Boris Aronov, Jacob E. Goodman, Richard Pollack

    Research output: Contribution to journalArticle

    Abstract

    We prove that a collection of compact convex sets of bounded diameters in K.d that is unbounded in k independent directions has a fc-flat transversal for k < d if and only if every d + 1 of the sets have a /:-transversal. This result generalizes a theorem of Hadwiger(-Danzer-Griinbaum-Klee) on line transversals for an unbounded family of compact convex sets. It is the first Helly-type theorem known for transversals of dimension between 1 and d -1.

    Original languageEnglish (US)
    Pages (from-to)177-183
    Number of pages7
    JournalComputational Geometry: Theory and Applications
    Volume21
    Issue number3
    StatePublished - 2002

    Fingerprint

    Helly-type Theorems
    Transversals
    Compact Convex Set
    High-dimensional
    If and only if
    Generalise
    Line
    Theorem

    Keywords

    • -4-unbounded
    • Convex sets
    • Geometric transversal theory
    • Helly-type theorem
    • Jt-transversal

    ASJC Scopus subject areas

    • Computational Theory and Mathematics
    • Discrete Mathematics and Combinatorics
    • Geometry and Topology

    Cite this

    Aronov, B., Goodman, J. E., & Pollack, R. (2002). A Helly-type theorem for higher-dimensional transversals. Computational Geometry: Theory and Applications, 21(3), 177-183.

    A Helly-type theorem for higher-dimensional transversals. / Aronov, Boris; Goodman, Jacob E.; Pollack, Richard.

    In: Computational Geometry: Theory and Applications, Vol. 21, No. 3, 2002, p. 177-183.

    Research output: Contribution to journalArticle

    Aronov, B, Goodman, JE & Pollack, R 2002, 'A Helly-type theorem for higher-dimensional transversals', Computational Geometry: Theory and Applications, vol. 21, no. 3, pp. 177-183.
    Aronov, Boris ; Goodman, Jacob E. ; Pollack, Richard. / A Helly-type theorem for higher-dimensional transversals. In: Computational Geometry: Theory and Applications. 2002 ; Vol. 21, No. 3. pp. 177-183.
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