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

Boris Aronov, Jacob E. Goodman, Richard Pollack, Rephael Wenger

    Research output: Chapter in Book/Report/Conference proceedingConference contribution

    Abstract

    Let S be a family of compact convex sets in Rd. Let D(S) be the largest diameter of any member of S. The family 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 ε/D(S) away from all d of the sets. We prove that if S is an ε-separated family 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)
    Title of host publicationProceedings of the Annual Symposium on Computational Geometry
    PublisherACM
    Pages57-63
    Number of pages7
    StatePublished - 2000
    Event16th Annual Symposium on Computational Geometry - Hong Kong, Hong Kong
    Duration: Jun 12 2000Jun 14 2000

    Other

    Other16th Annual Symposium on Computational Geometry
    CityHong Kong, Hong Kong
    Period6/12/006/14/00

    Fingerprint

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

    ASJC Scopus subject areas

    • Chemical Health and Safety
    • Software
    • Safety, Risk, Reliability and Quality
    • Geometry and Topology

    Cite this

    Aronov, B., Goodman, J. E., Pollack, R., & Wenger, R. (2000). Helly-type theorem for hyperplane transversals to well-separated convex sets. In Proceedings of the Annual Symposium on Computational Geometry (pp. 57-63). ACM.

    Helly-type theorem for hyperplane transversals to well-separated convex sets. / Aronov, Boris; Goodman, Jacob E.; Pollack, Richard; Wenger, Rephael.

    Proceedings of the Annual Symposium on Computational Geometry. ACM, 2000. p. 57-63.

    Research output: Chapter in Book/Report/Conference proceedingConference contribution

    Aronov, B, Goodman, JE, Pollack, R & Wenger, R 2000, Helly-type theorem for hyperplane transversals to well-separated convex sets. in Proceedings of the Annual Symposium on Computational Geometry. ACM, pp. 57-63, 16th Annual Symposium on Computational Geometry, Hong Kong, Hong Kong, 6/12/00.
    Aronov B, Goodman JE, Pollack R, Wenger R. Helly-type theorem for hyperplane transversals to well-separated convex sets. In Proceedings of the Annual Symposium on Computational Geometry. ACM. 2000. p. 57-63
    Aronov, Boris ; Goodman, Jacob E. ; Pollack, Richard ; Wenger, Rephael. / Helly-type theorem for hyperplane transversals to well-separated convex sets. Proceedings of the Annual Symposium on Computational Geometry. ACM, 2000. pp. 57-63
    @inproceedings{f521e451b2cd4587ac71e04b285e4165,
    title = "Helly-type theorem for hyperplane transversals to well-separated convex sets",
    abstract = "Let S be a family of compact convex sets in Rd. Let D(S) be the largest diameter of any member of S. The family S is ε-separated if, for every 0d 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, {Jacob E.} and Richard Pollack and Rephael Wenger",
    year = "2000",
    language = "English (US)",
    pages = "57--63",
    booktitle = "Proceedings of the Annual Symposium on Computational Geometry",
    publisher = "ACM",

    }

    TY - GEN

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

    AU - Aronov, Boris

    AU - Goodman, Jacob E.

    AU - Pollack, Richard

    AU - Wenger, Rephael

    PY - 2000

    Y1 - 2000

    N2 - Let S be a family of compact convex sets in Rd. Let D(S) be the largest diameter of any member of S. The family S is ε-separated if, for every 0d 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 family of compact convex sets in Rd. Let D(S) be the largest diameter of any member of S. The family S is ε-separated if, for every 0d 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=0033707561&partnerID=8YFLogxK

    UR - http://www.scopus.com/inward/citedby.url?scp=0033707561&partnerID=8YFLogxK

    M3 - Conference contribution

    SP - 57

    EP - 63

    BT - Proceedings of the Annual Symposium on Computational Geometry

    PB - ACM

    ER -