Distance k-sectors exist

Keiko Imai, Akitoshi Kawamura, Jiří Matoušek, Daniel Reem, Takeshi Tokuyama

Research output: Contribution to journalArticle

7 Citations (Scopus)

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 languageEnglish
Pages (from-to)713-720
Number of pages8
JournalComputational Geometry: Theory and Applications
Volume43
Issue number9
DOIs
Publication statusPublished - Nov 1 2010
Externally publishedYes

Fingerprint

Sector
Bisector
VLSI Design
Closed set
Geodesic
Metric space
Fixed point theorem
Euclidean space
Disjoint
Existence and Uniqueness
Uniqueness
Integer
Arbitrary

All Science Journal Classification (ASJC) codes

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

Cite this

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

In: Computational Geometry: Theory and Applications, Vol. 43, No. 9, 01.11.2010, p. 713-720.

Research output: Contribution to journalArticle

Imai, K, Kawamura, A, Matoušek, J, Reem, D & Tokuyama, T 2010, 'Distance k-sectors exist', Computational Geometry: Theory and Applications, vol. 43, no. 9, pp. 713-720. https://doi.org/10.1016/j.comgeo.2010.05.001
Imai, Keiko ; Kawamura, Akitoshi ; Matoušek, Jiří ; Reem, Daniel ; Tokuyama, Takeshi. / Distance k-sectors exist. In: Computational Geometry: Theory and Applications. 2010 ; Vol. 43, No. 9. pp. 713-720.
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