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 . We establish variance asymptotics as for the re-scaled intrinsic volumes and -face functionals of , , 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 with intensity . The scaling limit of the boundary of as 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.