On Point Processes Defined by Angular Conditions on Delaunay Neighbors in the Poisson-Voronoi Tessellation

Abstract

Consider a homogeneous Poisson point process of the Euclidean plane and its Voronoi tessellation. The present note discusses the properties of two stationary point processes associated with the latter and depending on a parameter $\theta$. The first one is the set of points that belong to some one-dimensional facet of the Voronoi tessellation and are such that the angle with which they see the two nuclei defining the facet is $\theta$. The main question of interest on this first point process is its intensity. The second point process is that of the intersections of the said tessellation with a straight line having a random orientation. Its intensity is well known. The intersection points almost surely belong to one-dimensional facets. The main question here is about the Palm distribution of the angle with which the points of this second point process see the two nuclei associated with the facet. The note gives answers to these two questions and briefly discusses their practical motivations. It also discusses natural extensions to dimension three.