### 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) |
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Pages (from-to) | 1300-1305 |

Number of pages | 6 |

Journal | Discrete Mathematics |

Volume | 339 |

Issue number | 4 |

DOIs | |

Publication status | Published - Apr 6 2016 |

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### 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