Consider an homogeneous PPP in $\mathbb{R}^2$ with intensity parameter $\lambda > 0$, and denote the set of generator seeds in the spatial domain as $P = \{p_1, p_2, \dots, p_N\}$. At the same time, envision a Voronoi tessellation of the planar point set such that the space is sparated into regions based on the proximity of points in the space to the seed points $p_i \in P$, as shown in the figure below.
In this setup, I am interested in obtaining relevant insights about the cumulative distribution function of the extreme distances in the Voronoi diagram. In essence, the extreme distances of a Voronoi region $V(p_i)$ correspond to the smallest and largest Euclidean distances between the generator seed $p_i$ and any of the vertices (see the blue and magenta arrows in the figure).
Figure: Max. and min. distance in Poisson Voronoi Tessellation
Initial thoughts
The distribution of the distance between a seed point and any of its vertices has been shown to be Gamma distributed [1], with probability distribution, $$F_R(r) = 1 - \big(1 + \pi r^2 \big)e^{-\pi r^2}.$$
From there, my initial thought was to use order statistics. However, the stochastic arrangement of seed points also introduces a variability in the number of vertices associated with each seed point. Moreover, even though the interior cells in the Voronoi diagram tend to have more vertices than exterior cells, the Voronoi regions with the highest number of vertices are spread throughout the entire space and do not concentrate at any particular location [2].
On top of that, the distribution of the number of vertices (which is the same as the distribution of the number of edges) is not known and most works simply approximate it as a Gamma distribution. The only thing for sure is that the expected number of vertices is 6 with probability 1.
1: K. A. Brakke, “Statistics of random plane voronoi tessellations,” Department of Mathematical Sciences, Susquehanna University (Manuscript 1987a), vol. 18, 1987.
2: Gezer, Fatih; Aykroyd, Robert G.; Barber, Stuart, Statistical properties of Poisson-Voronoi tessellation cells in bounded regions, ZBL07493332.