Variance asymptotics and scaling limits for Gaussian polytopes

Abstract

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 $$n \rightarrow \infty $$ for the re-scaled intrinsic volumes and $$k$$ -face functionals of $$K_n$$ , $$k \in \{0,1,\ldots ,d-1\}$$ , resolving an open problem (Weil and Wieacker, Handbook of Convex Geometry, vol. B, pp. 1391–1438. North-Holland/Elsevier, Amsterdam, 1993). Variance asymptotics are given in terms of functionals of germ-grain models having parabolic grains with apices at a Poisson point process on $$\mathbb {R}^{d-1} \times \mathbb {R}$$ with intensity $$e^h dh dv$$ . The scaling limit of the boundary of $$K_n$$ as $$n \rightarrow \infty $$ 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.