### Abstract

Assume we have k points in general position in the plane such that the ratio between the maximum distance of any pair of points to the minimum distance of any pair of points is at most α√k, for some positive constant α. We show that there exist at least βk^{1/4} of these points which are the vertices of a convex polygon, for some positive constant β=β(α). On the other hand, we show that for every fixed ε>0, if k>k(ε), then there is a set of k points in the plane for which the above ratio is at most 4√k, which does not contain a convex polygon of more than k^{1/3+ε} vertices.

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

Pages (from-to) | 245-251 |

Number of pages | 7 |

Journal | Discrete & Computational Geometry |

Volume | 4 |

Issue number | 1 |

DOIs | |

State | Published - Dec 1 1989 |

Externally published | Yes |

### All Science Journal Classification (ASJC) codes

- Theoretical Computer Science
- Geometry and Topology
- Discrete Mathematics and Combinatorics
- Computational Theory and Mathematics

## Fingerprint Dive into the research topics of 'The maximum size of a convex polygon in a restricted set of points in the plane'. Together they form a unique fingerprint.

## Cite this

Alon, N., Katchalski, M., & Pulleyblank, W. R. (1989). The maximum size of a convex polygon in a restricted set of points in the plane.

*Discrete & Computational Geometry*,*4*(1), 245-251. https://doi.org/10.1007/BF02187725