TY - GEN

T1 - Zone diagrams in Euclidean spaces and in other normed spaces

AU - Kawamura, Akitoshi

AU - Matoušek, Jiřŕ

AU - Tokuyama, Takeshi

PY - 2010/7/30

Y1 - 2010/7/30

N2 - 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.

AB - 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.

KW - Knaster-tarski fixed point theorem

KW - Zone diagrams

UR - http://www.scopus.com/inward/record.url?scp=77954656727&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=77954656727&partnerID=8YFLogxK

U2 - 10.1145/1810959.1810997

DO - 10.1145/1810959.1810997

M3 - Conference contribution

AN - SCOPUS:77954656727

SN - 9781450300162

T3 - Proceedings of the Annual Symposium on Computational Geometry

SP - 216

EP - 221

BT - Proceedings of the 26th Annual Symposium on Computational Geometry, SCG'10

T2 - 26th Annual Symposium on Computational Geometry, SoCG 2010

Y2 - 13 June 2010 through 16 June 2010

ER -