The distributions of the smallest disks containing the Poisson-Voronoi typical cell and the Crofton cell in the plane

Among the disks centered at a typical particle of the two-dimensional Poisson-Voronoi tessellation, let Rm be the radius of the largest included within the polygonal cell associated with that particle and RM be the radius of the smallest containing that polygonal cell. In this article, we obtain the joint distribution of Rm and RM. This result is derived from the covering properties of the circle due to Stevens, Siegel and Holst. The same method works for studying the Crofton cell associated with the Poisson line process in the plane. The computation of the conditional probabilities P{RM . r + s | Rm = r} reveals the circular property of the Poisson-Voronoi typical cells (as well as the Crofton cells) having a .large. in-disk.