Zone diagrams: Existence, uniqueness and algorithmic challenge

A zone diagram is a new variation of the classical notion of Voronoi diagram. Given points (sites) p₁, ⋯;, p_n in the plane, each p_i, is assigned a region R_i, but in contrast to the ordinary Voronoi diagrams, the union of the R_i has a nonempty complement, the neutral zone. The defining property is that each R_i consists of all x ∈ ℝ² that lie closer (non-strictly) to p_i than to the union of all the other R_j, 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.