A Helly-type theorem for hyperplane transversals to well-separated convex sets

Boris Aronov, J. E. Goodman, R. Pollack, R. Wenger

    Research output: Contribution to journalArticle

    Abstract

    Let S be a finite collection of compact convex sets in ℝd. Let D(S) be the largest diameter of any member of S. We say that the collection S is ε-separated if, for every 0 < k < d, any k of the sets can be separated from any other d - k of the sets by a hyperplane more than εD(S)/2 away from all d of the sets. We prove that if S is an ε-separated collection of at least N (ε) compact convex sets in ℝd and every 2d + 2 members of S are met by a hyperplane, then there is a hyperplane meeting all the members of S. The number N (ε) depends both on the dimension d and on the separation parameter ε. This is the first Helly-type theorem known for hyperplane transversals to compact convex sets of arbitrary shape in dimension greater than one.

    Original languageEnglish (US)
    Pages (from-to)507-517
    Number of pages11
    JournalDiscrete and Computational Geometry
    Volume25
    Issue number4
    StatePublished - 2001

    Fingerprint

    Helly-type Theorems
    Transversals
    Hyperplane
    Convex Sets
    Compact Convex Set
    Arbitrary

    ASJC Scopus subject areas

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

    Cite this

    Aronov, B., Goodman, J. E., Pollack, R., & Wenger, R. (2001). A Helly-type theorem for hyperplane transversals to well-separated convex sets. Discrete and Computational Geometry, 25(4), 507-517.

    A Helly-type theorem for hyperplane transversals to well-separated convex sets. / Aronov, Boris; Goodman, J. E.; Pollack, R.; Wenger, R.

    In: Discrete and Computational Geometry, Vol. 25, No. 4, 2001, p. 507-517.

    Research output: Contribution to journalArticle

    Aronov, B, Goodman, JE, Pollack, R & Wenger, R 2001, 'A Helly-type theorem for hyperplane transversals to well-separated convex sets', Discrete and Computational Geometry, vol. 25, no. 4, pp. 507-517.
    Aronov, Boris ; Goodman, J. E. ; Pollack, R. ; Wenger, R. / A Helly-type theorem for hyperplane transversals to well-separated convex sets. In: Discrete and Computational Geometry. 2001 ; Vol. 25, No. 4. pp. 507-517.
    @article{9f9ed48979354e3a9b6601731766ff29,
    title = "A Helly-type theorem for hyperplane transversals to well-separated convex sets",
    abstract = "Let S be a finite collection of compact convex sets in ℝd. Let D(S) be the largest diameter of any member of S. We say that the collection S is ε-separated if, for every 0 < k < d, any k of the sets can be separated from any other d - k of the sets by a hyperplane more than εD(S)/2 away from all d of the sets. We prove that if S is an ε-separated collection of at least N (ε) compact convex sets in ℝd and every 2d + 2 members of S are met by a hyperplane, then there is a hyperplane meeting all the members of S. The number N (ε) depends both on the dimension d and on the separation parameter ε. This is the first Helly-type theorem known for hyperplane transversals to compact convex sets of arbitrary shape in dimension greater than one.",
    author = "Boris Aronov and Goodman, {J. E.} and R. Pollack and R. Wenger",
    year = "2001",
    language = "English (US)",
    volume = "25",
    pages = "507--517",
    journal = "Discrete and Computational Geometry",
    issn = "0179-5376",
    publisher = "Springer New York",
    number = "4",

    }

    TY - JOUR

    T1 - A Helly-type theorem for hyperplane transversals to well-separated convex sets

    AU - Aronov, Boris

    AU - Goodman, J. E.

    AU - Pollack, R.

    AU - Wenger, R.

    PY - 2001

    Y1 - 2001

    N2 - Let S be a finite collection of compact convex sets in ℝd. Let D(S) be the largest diameter of any member of S. We say that the collection S is ε-separated if, for every 0 < k < d, any k of the sets can be separated from any other d - k of the sets by a hyperplane more than εD(S)/2 away from all d of the sets. We prove that if S is an ε-separated collection of at least N (ε) compact convex sets in ℝd and every 2d + 2 members of S are met by a hyperplane, then there is a hyperplane meeting all the members of S. The number N (ε) depends both on the dimension d and on the separation parameter ε. This is the first Helly-type theorem known for hyperplane transversals to compact convex sets of arbitrary shape in dimension greater than one.

    AB - Let S be a finite collection of compact convex sets in ℝd. Let D(S) be the largest diameter of any member of S. We say that the collection S is ε-separated if, for every 0 < k < d, any k of the sets can be separated from any other d - k of the sets by a hyperplane more than εD(S)/2 away from all d of the sets. We prove that if S is an ε-separated collection of at least N (ε) compact convex sets in ℝd and every 2d + 2 members of S are met by a hyperplane, then there is a hyperplane meeting all the members of S. The number N (ε) depends both on the dimension d and on the separation parameter ε. This is the first Helly-type theorem known for hyperplane transversals to compact convex sets of arbitrary shape in dimension greater than one.

    UR - http://www.scopus.com/inward/record.url?scp=0035617026&partnerID=8YFLogxK

    UR - http://www.scopus.com/inward/citedby.url?scp=0035617026&partnerID=8YFLogxK

    M3 - Article

    VL - 25

    SP - 507

    EP - 517

    JO - Discrete and Computational Geometry

    JF - Discrete and Computational Geometry

    SN - 0179-5376

    IS - 4

    ER -