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 2012

All Science Journal Classification (ASJC) codes

  • Mathematics(all)

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