### Abstract

We prove an optimal extension of the centerpoint theorem: given a set P of n points in the plane, there exist two points (not necessarily among input points) that hit all convex sets containing more than 47n points of P. We further prove that this bound is tight. We get this bound as part of a more general procedure for finding small number of points hitting convex sets over P, yielding several improvements over previous results.

Original language | English (US) |
---|---|

Pages (from-to) | 505-510 |

Number of pages | 6 |

Journal | Computational Geometry: Theory and Applications |

Volume | 42 |

Issue number | 6-7 |

DOIs | |

State | Published - Aug 1 2009 |

### Fingerprint

### Keywords

- Centerpoint theorem
- Combinatorial geometry
- Discrete geometry
- Extremal methods
- Hitting convex sets
- Weak -nets

### ASJC Scopus subject areas

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

### Cite this

*Computational Geometry: Theory and Applications*,

*42*(6-7), 505-510. https://doi.org/10.1016/j.comgeo.2007.10.004

**An optimal extension of the centerpoint theorem.** / Mustafa, Nabil H.; Ray, Saurabh.

Research output: Contribution to journal › Article

*Computational Geometry: Theory and Applications*, vol. 42, no. 6-7, pp. 505-510. https://doi.org/10.1016/j.comgeo.2007.10.004

}

TY - JOUR

T1 - An optimal extension of the centerpoint theorem

AU - Mustafa, Nabil H.

AU - Ray, Saurabh

PY - 2009/8/1

Y1 - 2009/8/1

N2 - We prove an optimal extension of the centerpoint theorem: given a set P of n points in the plane, there exist two points (not necessarily among input points) that hit all convex sets containing more than 47n points of P. We further prove that this bound is tight. We get this bound as part of a more general procedure for finding small number of points hitting convex sets over P, yielding several improvements over previous results.

AB - We prove an optimal extension of the centerpoint theorem: given a set P of n points in the plane, there exist two points (not necessarily among input points) that hit all convex sets containing more than 47n points of P. We further prove that this bound is tight. We get this bound as part of a more general procedure for finding small number of points hitting convex sets over P, yielding several improvements over previous results.

KW - Centerpoint theorem

KW - Combinatorial geometry

KW - Discrete geometry

KW - Extremal methods

KW - Hitting convex sets

KW - Weak -nets

UR - http://www.scopus.com/inward/record.url?scp=84867956198&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84867956198&partnerID=8YFLogxK

U2 - 10.1016/j.comgeo.2007.10.004

DO - 10.1016/j.comgeo.2007.10.004

M3 - Article

VL - 42

SP - 505

EP - 510

JO - Computational Geometry: Theory and Applications

JF - Computational Geometry: Theory and Applications

SN - 0925-7721

IS - 6-7

ER -