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 ${\mathbb{R}}^d$. We establish variance asymptotics as $n\to \mathrm{\infty }$ for the re-scaled intrinsic volumes and $k$-face functionals of $K_n$, $k\in \{0,1,\mathrm{.}\mathrm{.}\mathrm{.},d-1\}$, 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 $\R^{d-1} \times \R$withintensity$e^h dh dv$\mathrm{.}Thescalinglimitoftheboundaryof$K_n$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.