An optimal generalization of the Colorful Carathéodory theorem

Nabil H. Mustafa, Saurabh Ray

    Research output: Contribution to journalArticle

    Abstract

    The Colorful Carathéodory theorem by Bárány (1982) states that given d+1 sets of points in Rd, the convex hull of each containing the origin, there exists a simplex (called a 'rainbow simplex') with at most one point from each point set, which also contains the origin. Equivalently, either there is a hyperplane separating one of these d+1 sets of points from the origin, or there exists a rainbow simplex containing the origin. One of our results is the following extension of the Colorful Carathéodory theorem: given ⌊d/2⌋+1 sets of points in Rd and a convex object C, then either one set can be separated from C by a constant (depending only on d) number of hyperplanes, or there is a ⌊d/2⌋-dimensional rainbow simplex intersecting C.

    Original languageEnglish (US)
    Pages (from-to)1300-1305
    Number of pages6
    JournalDiscrete Mathematics
    Volume339
    Issue number4
    DOIs
    StatePublished - Apr 6 2016

    Fingerprint

    Set of points
    Theorem
    Hyperplane
    Convex Hull
    Point Sets
    Generalization
    Object

    Keywords

    • Carathéodory's theorem
    • Colorful Carathéodory's theorem
    • Convexity
    • Hadwiger-Debrunner (p,q) theorem and weak epsilon-nets
    • Separating hyperplanes

    ASJC Scopus subject areas

    • Theoretical Computer Science
    • Discrete Mathematics and Combinatorics

    Cite this

    An optimal generalization of the Colorful Carathéodory theorem. / Mustafa, Nabil H.; Ray, Saurabh.

    In: Discrete Mathematics, Vol. 339, No. 4, 06.04.2016, p. 1300-1305.

    Research output: Contribution to journalArticle

    Mustafa, Nabil H. ; Ray, Saurabh. / An optimal generalization of the Colorful Carathéodory theorem. In: Discrete Mathematics. 2016 ; Vol. 339, No. 4. pp. 1300-1305.
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