Integration of combinatorial decompositions in the presence of collinearities (Q946083)

Integration procedures concerning combinatorial integral geometry are often carried out with respect to the combinatorial decompositions associated to finite point sets in the Euclidean plane. Usually, the further assumption that no three points are collinear is considered. In the present paper the author drops such a requirement, and shows that this leads to connections with stochastic geometry. The focus is on two main problems. The first one concerns \textit{perforated convex domains}, namely convex domains containing a finite array of non-overlapping convex holes. The second question regards \textit{polygonal colorings}, defined to be integer-valued functions which are constant on polygons. In particular, polygonal colorings generated by random Poisson polygonal process are considered.