Zone diagrams in Euclidean spaces and in other normed spaces

Akitoshi Kawamura, Jiřŕ Matoušek, Takeshi Tokuyama

研究成果: Chapter in Book/Report/Conference proceedingConference contribution

7 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 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
ホスト出版物のタイトルProceedings of the 26th Annual Symposium on Computational Geometry, SCG'10
ページ216-221
ページ数6
DOI
出版ステータス出版済み - 2010
イベント26th Annual Symposium on Computational Geometry, SoCG 2010 - Snowbird, UT, 米国
継続期間: 6 13 20106 16 2010

出版物シリーズ

名前Proceedings of the Annual Symposium on Computational Geometry

その他

その他26th Annual Symposium on Computational Geometry, SoCG 2010
国/地域米国
CitySnowbird, UT
Period6/13/106/16/10

All Science Journal Classification (ASJC) codes

  • 理論的コンピュータサイエンス
  • 幾何学とトポロジー
  • 計算数学

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