Integration of combinatorial decompositions in the presence of collinearities

Abstract

Many results in Combinatorial Integral Geometry are derived by integration of the combinatorial decompositions associated with finite point sets {P i } given in the plane ℝ2. However, most previous cases of integration of the decompositions in question were carried out for the point sets {P i } containing no triads of collinear points, where the familiar algorithm sometimes called the “Four indicator formula” can be used. The present paper is to demonstrate that the complete combinatorial algorithm valid for sets {P i } not subject to the mentioned restriction opens the path to various results, including the field of Stochastic Geometry. In the paper the complete algorithm is applied first in an integration procedure in a study of the perforated convex domains, i.e convex domains containing a finite array of non-overlapping convex holes. The second application is in the study of random colorings of the plane that are Euclidean motions invariant in distribution, basing on the theory of random polygonal windows from the so-called Independent Angles (IA) class. The method is a direct averaging of the complete combinatorial decompositions written for colorings observed in polygonal windows from the IA class. The approach seems to be quite general, but promises to be especially effective for the random coloring generated by random Poisson polygon process governed by the Haar measure on the group of Euclidean motions of the plane, assuming that a point P ∈ ℝ2 is colored J if P is covered by exactly J polygons of the Poisson process. A general theorem clearing the way for Laplace transform treatment of the random colorings induced on line segments is formulated.