Distance from the Nucleus to a Uniformly Random Point in the Typical and the Crofton Cells of the Poisson-Voronoi Tessellation
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Consider the distances R_o and R̃_o from the nucleus to a uniformly random point in the typical and Crofton cells, respectively, of the d-dimensional Poisson-Voronoi (PV) tessellation. The main objective of this paper is to characterize the exact distributions of R_o and R̃_o. First, using the well-known relationship between the Crofton cell and the typical cell, we show that the random variable R̃_o is equivalent in distribution to the contact distance of the Poisson point process. Next, we derive a multi-integral expression for the exact distribution of R_o. Further, we derive a closed-form approximate expression for the distribution of R_o, which is the contact distribution with a mean corrected by a factor equal to the ratio of the mean volumes of the Crofton and typical cells. An additional outcome of our analysis is a direct proof of the well-known spherical property of the PV cells having a large inball.
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