Distance k-sectors exist

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

Research output: Chapter in Book/Report/Conference proceedingConference contribution

2 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 C i-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
Title of host publicationProceedings of the 26th Annual Symposium on Computational Geometry, SCG'10
Pages210-215
Number of pages6
DOIs
Publication statusPublished - Jul 30 2010
Externally publishedYes
Event26th Annual Symposium on Computational Geometry, SoCG 2010 - Snowbird, UT, United States
Duration: Jun 13 2010Jun 16 2010

Publication series

NameProceedings of the Annual Symposium on Computational Geometry

Other

Other26th Annual Symposium on Computational Geometry, SoCG 2010
CountryUnited States
CitySnowbird, UT
Period6/13/106/16/10

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All Science Journal Classification (ASJC) codes

  • Theoretical Computer Science
  • Geometry and Topology
  • Computational Mathematics

Cite this

Imai, K., Kawamura, A., Matoušek, J., Reem, D., & Tokuyama, T. (2010). Distance k-sectors exist. In 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