### 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 | |

State | Published - Apr 6 2016 |

### Fingerprint

### 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

*Discrete Mathematics*,

*339*(4), 1300-1305. https://doi.org/10.1016/j.disc.2015.11.019

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

Research output: Contribution to journal › Article

*Discrete Mathematics*, vol. 339, no. 4, pp. 1300-1305. https://doi.org/10.1016/j.disc.2015.11.019

}

TY - JOUR

T1 - An optimal generalization of the Colorful Carathéodory theorem

AU - Mustafa, Nabil H.

AU - Ray, Saurabh

PY - 2016/4/6

Y1 - 2016/4/6

N2 - 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.

AB - 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.

KW - Carathéodory's theorem

KW - Colorful Carathéodory's theorem

KW - Convexity

KW - Hadwiger-Debrunner (p,q) theorem and weak epsilon-nets

KW - Separating hyperplanes

UR - http://www.scopus.com/inward/record.url?scp=84949964704&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84949964704&partnerID=8YFLogxK

U2 - 10.1016/j.disc.2015.11.019

DO - 10.1016/j.disc.2015.11.019

M3 - Article

VL - 339

SP - 1300

EP - 1305

JO - Discrete Mathematics

JF - Discrete Mathematics

SN - 0012-365X

IS - 4

ER -