### 抜粋

The bisector of two nonempty sets P and Q in Rd 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 (C1,C2,...,Ck-1) such that Ci is the bisector of Ci-1 and Ci+1 for every i=1,2,...,k-1, where C0=P and Ck=Q. This notion, for the case where P and Q are points in R2, 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.

元の言語 | 英語 |
---|---|

ページ（範囲） | 713-720 |

ページ数 | 8 |

ジャーナル | Computational Geometry: Theory and Applications |

巻 | 43 |

発行部数 | 9 |

DOI | |

出版物ステータス | 出版済み - 11 1 2010 |

### フィンガープリント

### All Science Journal Classification (ASJC) codes

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

### これを引用

*Computational Geometry: Theory and Applications*,

*43*(9), 713-720. https://doi.org/10.1016/j.comgeo.2010.05.001