Mathematische Annalen

, Volume 354, Issue 4, pp 1201–1221 | Cite as

Zone diagrams in Euclidean spaces and in other normed spaces

  • Akitoshi Kawamura
  • Jiří Matoušek
  • Takeshi Tokuyama


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.


Euclidean Space Unit Ball Normed Space Triangle Inequality Voronoi Diagram 
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Copyright information

© Springer-Verlag 2011

Authors and Affiliations

  • Akitoshi Kawamura
    • 1
  • Jiří Matoušek
    • 2
    • 3
  • Takeshi Tokuyama
    • 4
  1. 1.Department of Computer ScienceUniversity of TokyoTokyoJapan
  2. 2.Department of Applied Mathematics, Institute of Theoretical Computer Science (ITI)Charles UniversityPraha 1Czech Republic
  3. 3.Institute of Theoretical Computer ScienceETH ZürichZürichSwitzerland
  4. 4.Graduate School of Information SciencesTohoku UniversitySendaiJapan

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