### Abstract

The bisector of two nonempty sets P and Q in ℝ^{d} is the set of all points with equal distance to P and to Q. A distance k-sector of P and Q, where k ≥ 2 is an integer, is a (k -1)-tuple (C_{1},C_{2}, . . . ,C_{k-1}) such that C_{i} is the bisector of C _{i-1} and C_{i+1} for every i = 1, 2, . . . , k - 1, where C_{0} = P and C_{k} = Q. This notion, for the case where P and Q are points in ℝ^{2}, was introduced by Asano, Matoušek, and Tokuyama, motivated by a question of Murata in VLSI design. They established the existence and uniqueness of the distance 3-sector in this special case. We prove the existence of a distance k-sector for all k and for every two disjoint, nonempty, closed sets P and Q in Euclidean spaces of any (finite) dimension (uniqueness remains open), or more generally, in proper geodesic spaces. The core of the proof is a new notion of k-gradation for P and Q, whose existence (even in an arbitrary metric space) is proved using the Knaster-Tarski fixed point theorem, by a method introduced by Reem and Reich for a slightly different purpose.

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

Title of host publication | Proceedings of the 26th Annual Symposium on Computational Geometry, SCG'10 |

Pages | 210-215 |

Number of pages | 6 |

DOIs | |

Publication status | Published - Jul 30 2010 |

Externally published | Yes |

Event | 26th Annual Symposium on Computational Geometry, SoCG 2010 - Snowbird, UT, United States Duration: Jun 13 2010 → Jun 16 2010 |

### Publication series

Name | Proceedings of the Annual Symposium on Computational Geometry |
---|

### Other

Other | 26th Annual Symposium on Computational Geometry, SoCG 2010 |
---|---|

Country | United States |

City | Snowbird, UT |

Period | 6/13/10 → 6/16/10 |

### Fingerprint

### All Science Journal Classification (ASJC) codes

- Theoretical Computer Science
- Geometry and Topology
- Computational Mathematics

### Cite this

*Proceedings of the 26th Annual Symposium on Computational Geometry, SCG'10*(pp. 210-215). (Proceedings of the Annual Symposium on Computational Geometry). https://doi.org/10.1145/1810959.1810996