Distance k-sectors exist

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

Research output: Contribution to journalArticlepeer-review

8 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 2010

All Science Journal Classification (ASJC) codes

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

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