Abstract
The study of distribution functions (with respect to areas, perimeters) for partitioning a plane (space) by a random field of straight lines (hyperplanes) and for obtaining Voronoi diagrams is a classical problem in statistical geometry. Moments for such distributions have been investigated since 1972 [1]. We give a complete solution of these problems for the plane, as well as for Voronoi diagrams. The following problems are solved: 1. A random set of straight lines is given on the plane, all shifts are equiprobable, and the distribution law has the form \(F(\varphi ).\) What is the area (perimeter) distribution of the parts of the partition? 2. A random set of points is marked on the plane. Each point A is associated with a “region of attraction,” which is a set of points on the plane to which A is the closest of the marked set. The idea is to interpret a random polygon as the evolution of a segment on a moving one and construct kinetic equations. It is sufficient to take into account a limited number of parameters: the covered area (perimeter), the length of the segment, and the angles at its ends. We show how to reduce these equations to the Riccati equation using the Laplace transform.
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REFERENCES
R. E. Miles, “The random division of space,” Adv. Appl. Probab. 4, 243–266 (1972).
A. Ya. Belov, “Statistical geometry and equilibrium of block arrays,” Candidate’s Dissertation in Physics and Mathematics (Moscow State University, Moscow, 1991).
R. E. Miles, “Poisson flats in Euclidean spaces,” Adv. Appl. Probab. 1, 211–237 (1969).
M. G. Kendall and P. A. Moran, Geometric Probability (Hafner, New York, 1961).
A. Ya. Belov, “About random partitions,” Available from VINITI (VINITI, Moscow, 1991), No. 273-B91.
A. Kanel-Belov, M. Golafshan, S. Malev, and R. Yavich, “About random splitting of the plane,” Crimean Autumn Mathematical School-Symposium, KROMSH (2020), pp. 294–295.
Z. Kabluchko, “Angles of random simplices and face numbers of random polytopes,” Adv. Math. 380, 107612 (2021).
P. Calka, “An explicit expression for the distribution of the number of sides of the typical Poisson–Voronoi cell,” Adv. Appl. Probab. 35 (4), 863–870 (2003).
P. Calka, “Precise formulae for the distributions of the principal geometric characteristics of the typical cells of a two-dimensional Poisson–Voronoi tessellation and a Poisson line process,” Adv. Appl. Probab. 35 (3), 551–562 (2016).
L. A. Centaló, Integral Geometry and Geometric Probabilities (Cambridge Univ. Press, Cambridge, 2009).
R. V. Ambartsumyan, J. Mekke, and D. Shtoyan, Introduction to Stochastic Geometry (Nauka, Moscow, 1989) [in Russian].
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This work was supported by the Russian Science Foundation, grant no. 22-19-20073.
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Translated by I. Ruzanova
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Kanel-Belov, A.Y., Golafshan, M., Malev, S.G. et al. Finding the Area and Perimeter Distributions for Flat Poisson Processes of a Straight Line and Voronoi Diagrams. Dokl. Math. 109, 56–61 (2024). https://doi.org/10.1134/S1064562424701801
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DOI: https://doi.org/10.1134/S1064562424701801