Variance Asymptotics and Scaling Limits for Gaussian Polytopes

Let Kn be the convex hull of i.i.d. random variables distributed according to the standard normal distribution on Rd. We establish variance asymptotics as n → ∞ for the re-scaled intrinsic volumes and k-face functionals of Kn, 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 Rd−1 × R with intensity ehdhdv. The scaling limit of the boundary of Kn 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. 1 Main results For all λ ∈ [1,∞), let Pλ denote a Poisson point process of intensity λφ(x)dx, where φ(x): = (2pi)−d/2 exp(−|x