### Abstract

It is an old problem of Danzer and Rogers to decide whether it is possible to arrange O(1ε) points in the unit square so that every rectangle of area ε > 0 within the unit square contains at least one of them. We show that the answer to this question is in the negative if we slightly relax the notion of rectangles, as follows. Let δ be a fixed small positive number. A quasi-rectangle is a region swept out by a continuously moving segment s, with no rotation, so that throughout the motion the angle between the trajectory of the center of s and its normal vector remains at most δ. We show that the smallest number of points needed to pierce all quasi-rectangles of area ε > 0 within the unit square is Θ(1εlog1ε).

Original language | English |
---|---|

Pages (from-to) | 1391-1397 |

Number of pages | 7 |

Journal | Journal of Combinatorial Theory. Series A |

Volume | 119 |

Issue number | 7 |

DOIs | |

Publication status | Published - Oct 1 2012 |

### Keywords

- Danzer-Rogers problem
- Epsilon-nets
- Piercing
- Quasi-rectangles

### ASJC Scopus subject areas

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

## Fingerprint Dive into the research topics of 'Piercing quasi-rectangles-On a problem of Danzer and Rogers'. Together they form a unique fingerprint.

## Cite this

*Journal of Combinatorial Theory. Series A*,

*119*(7), 1391-1397. https://doi.org/10.1016/j.jcta.2012.03.011