Variance asymptotics and scaling limits for Gaussian Polytopes

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 $n to infty$ for the re-scaled intrinsic volumes and $k$-face functionals of $K_n$, $k in {0,1,...,d-1}$, resolving an open problem. Variance asymptotics are given in terms of functionals of germ-grain models having parabolic grains with apices at a Poisson point process on $R^{d-1} times R$ with intensity $e^h dh dv$. The scaling limit of the boundary of $K_n$ as $n to infty$ 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.