The basic structures of Voronoi and generalized Voronoi polygons

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Abstract
For each particle in an aggregate of point particles in the plane, the set of points having it as closest particle is a convex polygon, and the aggregate V of such Voronoi polygons tessellates the plane. The geometric and stochastic structure of a random Voronoi polygon relative to a homogeneous Poisson process is specified. Similarly, those points of the plane possessing the same n nearest particles constitute a convex polygon cell in the generalized Voronoi tessellation 𝒱 (n = 2, 3, ·· ·). In fact, 𝒱 = 𝒱1, but to ease exposition n always takes the values 2, 3, ···. A key geometrical lemma elucidates the geometric structure of members of 𝒱 n , showing it to be simpler in one important respect than that of members of 𝒱; in that, for each such N-gon of given ‘type', there is a uniquely determined set of N generating particles. The corresponding jacobian is given, and used to derive the basic ergodic structure of 𝒱 n relative to a homogeneous Poisson process. Unlike 𝒱 no 𝒱 n contains any triangles. As n →∞, the vertices of the quadrangles of 𝒱 n tend to circularity, so that the sums of their opposite interior angles tend to π.

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