2007/01/01 by Tetsuo Asano, Jiřı́ Matoušek, Takeshi Tokuyama · 3 citations
Computer Science · Engineering · #Advanced Graph Theory Research #Advanced Numerical Analysis Techniques #Computational Geometry and Mesh Generation
paper · doi:10.1137/06067095x
openalex publication_date 2007/01/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/06/11
A zone diagram is a new variation of the classical notion of the Voronoi diagram. Given points (sites) \mathbf p1,…,\mathbf pn in the plane, each \mathbf pi is assigned a region Ri, but in contrast to the ordinary Voronoi diagrams, the union of the Ri has a nonempty complement, the neutral zone. The defining property is that each Ri consists of all \mathbf x∈ℝ2 that lie closer (nonstrictly) to \mathbf pi than to the union of all the other Rj, j≠ i. Thus, the zone diagram is defined implicitly, by a “fixed-point property,” and neither its existence nor its uniqueness seem obvious. We establish existence using a general fixed-point result (a consequence of Schauder's theorem or Kakutani's theorem); this proof should generalize easily to related settings, say higher dimensions. Then we prove uniqueness of the zone diagram, as well as convergence of a natural iterative algorithm for computing it, by a geometric argument, which also relies on a result for the case of two sites in an earlier paper. Many challenging questions remain open.