### Abstract

It is an old problem of Danzer and Rogers to decide whether it is possible arrange O(1/ε) points in the unit square so that every rectangle of area ε 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 is Θ (1/ε log 1/ε).

Original language | English |
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Title of host publication | Algorithms and Data Structures - 12th International Symposium, WADS 2011, Proceedings |

Publisher | Springer Verlag |

Number of pages | 1 |

ISBN (Print) | 9783642222993 |

DOIs | |

Publication status | Published - Jan 1 2011 |

Event | 12th International Symposium on Algorithms and Data Structures, WADS 2011 - New York, United States Duration: Aug 15 2011 → Aug 17 2011 |

### Publication series

Name | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |
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Volume | 6844 LNCS |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Other

Other | 12th International Symposium on Algorithms and Data Structures, WADS 2011 |
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Country | United States |

City | New York |

Period | 8/15/11 → 8/17/11 |

### Fingerprint

### ASJC Scopus subject areas

- Theoretical Computer Science
- Computer Science(all)

### Cite this

*Algorithms and Data Structures - 12th International Symposium, WADS 2011, Proceedings*(Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 6844 LNCS). Springer Verlag. https://doi.org/10.1007/978-3-642-22300-6_55