An optimal generalization of the Colorful Carathéodory theorem

Nabil H. Mustafa; Saurabh Ray

doi:10.1016/j.disc.2015.11.019

An optimal generalization of the Colorful Carathéodory theorem

Nabil H. Mustafa, Saurabh Ray

Computer Science

Research output: Contribution to journal › Article › peer-review

Abstract

The Colorful Carathéodory theorem by Bárány (1982) states that given d+1 sets of points in R^d, 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 R^d 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 language	English (US)
Pages (from-to)	1300-1305
Number of pages	6
Journal	Discrete Mathematics
Volume	339
Issue number	4
DOIs	https://doi.org/10.1016/j.disc.2015.11.019
State	Published - Apr 6 2016

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

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10.1016/j.disc.2015.11.019

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title = "An optimal generalization of the Colorful Carath{\'e}odory theorem",

abstract = "The Colorful Carath{\'e}odory theorem by B{\'a}r{\'a}ny (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 Carath{\'e}odory 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.",

keywords = "Carath{\'e}odory's theorem, Colorful Carath{\'e}odory's theorem, Convexity, Hadwiger-Debrunner (p,q) theorem and weak epsilon-nets, Separating hyperplanes",

author = "Mustafa, {Nabil H.} and Saurabh Ray",

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year = "2016",

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doi = "10.1016/j.disc.2015.11.019",

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AU - Ray, Saurabh

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An optimal generalization of the Colorful Carathéodory theorem

Abstract

Keywords

ASJC Scopus subject areas

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