Maximal Zone Diagrams and their Computation

The notion of the zone diagram of a finite set of points in the Euclidean plane is an interesting and rich variation of the classical Voronoi diagram, introduced by Asano, Matousek, Tokuyama. Here, we define the more inclusive notion of a maximal zone diagram. The proof of existence of maximal zone diagrams depends on less restrictive initial conditions and is thus conveniently established via Zorn's lemma in contrast to the use of fixed-point theory in proving the existence of a unique zone diagram. A zone diagram is a particular maximal zone diagram satisfying a unique dominance property. We give a characterization for maximal zone diagrams which allows recognition of maximality of certain subsets called subzone diagrams, as well as that of their iterative improvement toward maximality. Maximal zone diagrams offer their own interesting theoretical and computational challenges.