Zone diagrams in Euclidean spaces and in other normed spaces

Akitoshi Kawamura, Jiří Matoušek, Takeshi Tokuyama

研究成果: Contribution to journalArticle査読

4 被引用数 (Scopus)

抄録

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.

本文言語英語
ページ(範囲)1201-1221
ページ数21
ジャーナルMathematische Annalen
354
4
DOI
出版ステータス出版済み - 11 2012

All Science Journal Classification (ASJC) codes

  • 数学 (全般)

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