Zone diagrams in Euclidean spaces and in other normed spaces

Akitoshi Kawamura, Jiří Matoušek, Takeshi Tokuyama

Research output: Contribution to journalArticle

4 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 the 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
Pages (from-to)1201-1221
Number of pages21
JournalMathematische Annalen
Volume354
Issue number4
DOIs
Publication statusPublished - Nov 1 2012

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

  • Mathematics(all)

Cite this

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

In: Mathematische Annalen, Vol. 354, No. 4, 01.11.2012, p. 1201-1221.

Research output: Contribution to journalArticle

Kawamura, Akitoshi ; Matoušek, Jiří ; Tokuyama, Takeshi. / Zone diagrams in Euclidean spaces and in other normed spaces. In: Mathematische Annalen. 2012 ; Vol. 354, No. 4. pp. 1201-1221.
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