Some theoretical challenges in digital geometry: A perspective
Discrete Applied Mathematics
Proceedings of the twenty-sixth annual symposium on Computational geometry
Zone diagrams in Euclidean spaces and in other normed spaces
Proceedings of the twenty-sixth annual symposium on Computational geometry
Computational Geometry: Theory and Applications
The Geometric Stability of Voronoi Diagrams with Respect to Small Changes of the Sites
Proceedings of the twenty-seventh annual symposium on Computational geometry
Mollified zone diagrams and their computation
Transactions on Computational Science XIV
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A zone diagram is a new variation of the classical notion of the Voronoi diagram. Given points (sites) ${\mathbf p}_1,\ldots,{\mathbf p}_n$ in the plane, each ${\mathbf p}_i$ 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 ${\mathbf x}\in{\mathbb{R}}^2$ that lie closer (nonstrictly) to ${\mathbf p}_i$ than to the union of all the other $R_j$, $j\ne i$. 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.