Voronoi diagrams—a survey of a fundamental geometric data structure
ACM Computing Surveys (CSUR)
Spatial tessellations: concepts and applications of Voronoi diagrams
Spatial tessellations: concepts and applications of Voronoi diagrams
Distance Trisector Curves in Regular Convex Distance Me
ISVD '06 Proceedings of the 3rd International Symposium on Voronoi Diagrams in Science and Engineering
Distance Trisector of Segments and Zone Diagram of Segments in a Plane
ISVD '07 Proceedings of the 4th International Symposium on Voronoi Diagrams in Science and Engineering
Zone Diagrams: Existence, Uniqueness, and Algorithmic Challenge
SIAM Journal on Computing
Complexity theory for operators in analysis
Proceedings of the forty-second ACM symposium on Theory of computing
Proceedings of the twenty-sixth annual symposium on Computational geometry
Computational Geometry: Theory and Applications
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Zone diagram is a variation on the classical concept of a Voronoi diagram. 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, Matousek, 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.