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 - Jul 30 2010
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

Fingerprint

Normed Space
Euclidean space
Existence and Uniqueness
Diagram
Norm
Metric space
Arbitrary
Euclidean plane
Voronoi Diagram
Nonuniqueness
Equilibrium State
Disjoint
Fixed point

All Science Journal Classification (ASJC) codes

  • Theoretical Computer Science
  • Geometry and Topology
  • Computational Mathematics

Cite this

Kawamura, A., Matoušek, J., & Tokuyama, T. (2010). Zone diagrams in Euclidean spaces and in other normed spaces. In Proceedings of the 26th Annual Symposium on Computational Geometry, SCG'10 (pp. 216-221). (Proceedings of the Annual Symposium on Computational Geometry). https://doi.org/10.1145/1810959.1810997

Zone diagrams in Euclidean spaces and in other normed spaces. / Kawamura, Akitoshi; Matoušek, Jiřŕ; Tokuyama, Takeshi.

Proceedings of the 26th Annual Symposium on Computational Geometry, SCG'10. 2010. p. 216-221 (Proceedings of the Annual Symposium on Computational Geometry).

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

Kawamura, A, Matoušek, J & Tokuyama, T 2010, Zone diagrams in Euclidean spaces and in other normed spaces. in Proceedings of the 26th Annual Symposium on Computational Geometry, SCG'10. Proceedings of the Annual Symposium on Computational Geometry, pp. 216-221, 26th Annual Symposium on Computational Geometry, SoCG 2010, Snowbird, UT, United States, 6/13/10. https://doi.org/10.1145/1810959.1810997
Kawamura A, Matoušek J, Tokuyama T. Zone diagrams in Euclidean spaces and in other normed spaces. In Proceedings of the 26th Annual Symposium on Computational Geometry, SCG'10. 2010. p. 216-221. (Proceedings of the Annual Symposium on Computational Geometry). https://doi.org/10.1145/1810959.1810997
Kawamura, Akitoshi ; Matoušek, Jiřŕ ; Tokuyama, Takeshi. / Zone diagrams in Euclidean spaces and in other normed spaces. Proceedings of the 26th Annual Symposium on Computational Geometry, SCG'10. 2010. pp. 216-221 (Proceedings of the Annual Symposium on Computational Geometry).
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