## Abstract

Using a technique that Tverberg and Vrecica (1993) [16] discovered to give a surprisingly simple proof of Tverberg?s theorem, we show the following extension of the centerpoint theorem. Given any set P of n points in the plane, and a parameter 1/3 ≤ c ≤ 1, one can always find a disk D such that any closed half-space containing D contains at least cn points of P. Furthermore, D contains at most (3c ? 1)n/2 points of P (the case c = 1 is trivial ? take any D containing P; the case c = 1/3 is the centerpoint theorem). We also show that, for all c, this bound is tight up to a constant factor. We extend the upper bound to R^{d}. Specifically, we show that given any set P of n points, one can find a ball D containing at most ((d + 1)c ? 1)n/d points of P such that any half-space containing D contains at least cn points of P.

Original language | English |
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Pages (from-to) | 593-600 |

Number of pages | 8 |

Journal | Computational Geometry: Theory and Applications |

Volume | 43 |

Issue number | 6-7 |

DOIs | |

Publication status | Published - 1 Aug 2010 |

Externally published | Yes |

## Keywords

- Centerdisks
- Centerpoint theorem
- Data depth
- Discrete geometry
- Tverberg's theorem