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We study the Voronoi diagrams of a finite set of Cauchy distributions andtheir dual complexes from the viewpoint of information geometry by consideringthe Fisher-Rao distance, the Kullback-Leibler divergence, the chi squaredivergence, and a flat divergence derived from Tsallis' quadratic entropyrelated to the conformal flattening of the Fisher-Rao curved geometry. We provethat the Voronoi diagrams of the Fisher-Rao distance, the chi squaredivergence, and the Kullback-Leibler divergences all coincide with a hyperbolicVoronoi diagram on the corresponding Cauchy location-scale parameters, and thatthe dual Cauchy hyperbolic Delaunay complexes are Fisher orthogonal to theCauchy hyperbolic Voronoi diagrams. The dual Voronoi diagrams with respect tothe dual forward/reverse flat divergences amount to dual Bregman Voronoidiagrams, and their dual complexes are regular triangulations. The primalBregman-Tsallis Voronoi diagram corresponds to the hyperbolic Voronoi diagramand the dual Bregman-Tsallis Voronoi diagram coincides with the ordinaryEuclidean Voronoi diagram. Besides, we prove that the square root of theKullback-Leibler divergence between Cauchy distributions yields a metricdistance which is Hilbertian for the Cauchy scale families.

On Voronoi diagrams and dual Delaunay complexes on the information-geometric Cauchy manifolds