### Abstract

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.

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

Pages (from-to) | 713-720 |

Number of pages | 8 |

Journal | Computational Geometry: Theory and Applications |

Volume | 43 |

Issue number | 9 |

DOIs | |

Publication status | Published - Nov 1 2010 |

Externally published | Yes |

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### All Science Journal Classification (ASJC) codes

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

### Cite this

*Computational Geometry: Theory and Applications*,

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

**Distance k-sectors exist.** / Imai, Keiko; Kawamura, Akitoshi; Matoušek, Jiří; Reem, Daniel; Tokuyama, Takeshi.

Research output: Contribution to journal › Article

*Computational Geometry: Theory and Applications*, vol. 43, no. 9, pp. 713-720. https://doi.org/10.1016/j.comgeo.2010.05.001

}

TY - JOUR

T1 - Distance k-sectors exist

AU - Imai, Keiko

AU - Kawamura, Akitoshi

AU - Matoušek, Jiří

AU - Reem, Daniel

AU - Tokuyama, Takeshi

PY - 2010/11/1

Y1 - 2010/11/1

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=77954657071&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=77954657071&partnerID=8YFLogxK

U2 - 10.1016/j.comgeo.2010.05.001

DO - 10.1016/j.comgeo.2010.05.001

M3 - Article

AN - SCOPUS:77954657071

VL - 43

SP - 713

EP - 720

JO - Computational Geometry: Theory and Applications

JF - Computational Geometry: Theory and Applications

SN - 0925-7721

IS - 9

ER -