Zone diagrams in Euclidean spaces and in other normed spaces

Akitoshi Kawamura, Jiřŕ Matoušek, Takeshi Tokuyama

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

7 Citations (Scopus)

Abstract

Zone diagrams are a variation on the classical concept of Voronoi diagrams. Given n sites in a metric space that compete for territory, the zone diagram is an equilibrium state in the competition. Formally it is defined as a fixed point of a certain "dominance" map. Asano, Matoušek, and Tokuyama proved the existence and uniqueness of a zone diagram for point sites in Euclidean plane, and Reem and Reich showed existence for two arbitrary sites in an arbitrary metric space. We establish existence and uniqueness for n disjoint compact sites in a Euclidean space of arbitrary (finite) dimension, and more generally, in a finite-dimensional normed space with a smooth and rotund norm. The proof is considerably simpler than that of Asano et al. We also provide an example of non-uniqueness for a norm that is rotund but not smooth. Finally, we prove existence and uniqueness for two point sites in the plane with a smooth (but not necessarily rotund) norm.

Original languageEnglish
Title of host publicationProceedings of the 26th Annual Symposium on Computational Geometry, SCG'10
Pages216-221
Number of pages6
DOIs
Publication statusPublished - 2010 Jul 30
Event26th Annual Symposium on Computational Geometry, SoCG 2010 - Snowbird, UT, United States
Duration: 2010 Jun 132010 Jun 16

Publication series

NameProceedings of the Annual Symposium on Computational Geometry

Other

Other26th Annual Symposium on Computational Geometry, SoCG 2010
CountryUnited States
CitySnowbird, UT
Period10/6/1310/6/16

Keywords

  • Knaster-tarski fixed point theorem
  • Zone diagrams

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Geometry and Topology
  • Computational Mathematics

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