Volume degeneracy of the typical cell and the chord length distribution for Poisson-Voronoi tessellations in high dimensions

This paper is devoted to the study of some asymptotic behaviors of Poisson-Voronoi tessellation in the Euclidean space as the space dimension tends to .. We consider a family of homogeneous Poisson-Voronoi tessellations with constant intensity . in Euclidean spaces of dimensions n = 1, 2, 3, .. First we use the Blaschke-Petkantschin formula to prove that the variance of the volume of the typical cell tends to 0 exponentially in dimension. It is also shown that the volume of intersection of the typical cell with the co-centered ball of volume u converges in distribution to the constant ..1(1 . e..u). Next we consider the linear contact distribution function of the Poisson-Voronoi tessellation and compute the limit when the space dimension goes to .. As a by-product, the chord length distribution and the geometric covariogram of the typical cell are obtained in the limit.