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- W2016619947 abstract "A zone diagram is a new variation of the classical notion of Voronoi diagram. Given points (sites) P1,...,Pn in the plane, each 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 x ∈ R2 that lie closer (non-strictly) to Pi. than to the union of all the other Rj, j ≠ i. Thus, the zone diagram is defined implicitly, by a 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." @default.
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- W2016619947 date "2007-01-07" @default.
- W2016619947 modified "2023-09-26" @default.
- W2016619947 title "Zone diagrams: existence, uniqueness and algorithmic challenge" @default.
- W2016619947 doi "https://doi.org/10.5555/1283383.1283464" @default.
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