Schreiber and Yukich [Ann. Probab. 36 (2008) establish an asymptotic representation for random convex polytope geometry in the unit ball B d , d ≥ 2, in terms of the general theory of stabilizing functionals of Poisson point processes as well as in terms of generalized paraboloid growth processes. This paper further exploits this connection, introducing also a dual object termed the paraboloid hull process. Via these growth processes we establish local functional limit theorems for the properly scaled radius-vector and support functions of convex polytopes generated by high-density Poisson samples. We show that direct methods lead to explicit asymptotic expressions for the fidis of the properly scaled radius-vector and support functions. Generalized paraboloid growth processes, coupled with general techniques of stabilization theory, yield Brownian sheet limits for the defect volume and mean width functionals. Finally we provide explicit variance asymptotics and central limit theorems for the k-face and intrinsic volume functionals.

Let K n be the convex hull of i.i.d. random variables distributed according to the standard normal distribution on R d . We establish variance asymptotics as n → ∞ for the re-scaled intrinsic volumes and k-face functionals of K n , k ∈ {0, 1, ..., d − 1}, resolving an open problem [27]. Variance asymptotics are given in terms of functionals of germ-grain models having parabolic grains with apices at a Poisson point process on R d−1 × R with intensity e h dhdv. The scaling limit of the boundary of K n as n → ∞ converges to a festoon of parabolic surfaces, coinciding with that featuring in the geometric construction of the zero viscosity solution to Burgers' equation with random input.American Mathematical Society 2010 subject classifications. Primary 60F05, 52A20; Secondary 60D05, 52A23

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.

A homogeneous Poisson-Voronoi tessellation of intensity γ is observed in a convex body W . We associate to each cell of the tessellation two characteristic radii: the inradius, i.e. the radius of the largest ball centered at the nucleus and included in the cell, and the circumscribed radius, i.e. the radius of the smallest ball centered at the nucleus and containing the cell. We investigate the maximum and minimum of these two radii over all cells with nucleus in W . We prove that when γ → ∞, these four quantities converge to Gumbel or Weibull distributions up to a rescaling. Moreover, the contribution of boundary cells is shown to be negligible. Such approach is motivated by the analysis of the global regularity of the tessellation. In particular, consequences of our study include the convergence to the simplex shape of the cell with smallest circumscribed radius and an upper-bound for the Hausdorff distance between W and its so-called Poisson-Voronoi approximation.