### 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 (US) |
---|---|

Pages (from-to) | 593-600 |

Number of pages | 8 |

Journal | Computational Geometry: Theory and Applications |

Volume | 43 |

Issue number | 6-7 |

DOIs | |

State | Published - Aug 1 2010 |

### Fingerprint

### Keywords

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

### ASJC Scopus subject areas

- Computer Science Applications
- Geometry and Topology
- Control and Optimization
- Computational Theory and Mathematics
- Computational Mathematics

### Cite this

*Computational Geometry: Theory and Applications*,

*43*(6-7), 593-600. https://doi.org/10.1016/j.comgeo.2010.03.002

**Centerpoints and tverberg's technique.** / Basit, Abdul; Mustafa, Nabil H.; Ray, Saurabh; Raza, Sarfraz.

Research output: Contribution to journal › Article

*Computational Geometry: Theory and Applications*, vol. 43, no. 6-7, pp. 593-600. https://doi.org/10.1016/j.comgeo.2010.03.002

}

TY - JOUR

T1 - Centerpoints and tverberg's technique

AU - Basit, Abdul

AU - Mustafa, Nabil H.

AU - Ray, Saurabh

AU - Raza, Sarfraz

PY - 2010/8/1

Y1 - 2010/8/1

N2 - 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 Rd. 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.

AB - 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 Rd. 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.

KW - Centerdisks

KW - Centerpoint theorem

KW - Data depth

KW - Discrete geometry

KW - Tverberg's theorem

UR - http://www.scopus.com/inward/record.url?scp=77951880661&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=77951880661&partnerID=8YFLogxK

U2 - 10.1016/j.comgeo.2010.03.002

DO - 10.1016/j.comgeo.2010.03.002

M3 - Article

AN - SCOPUS:77951880661

VL - 43

SP - 593

EP - 600

JO - Computational Geometry: Theory and Applications

JF - Computational Geometry: Theory and Applications

SN - 0925-7721

IS - 6-7

ER -