# Variance asymptotics and scaling limits for Gaussian Polytopes

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1 LMRS - Laboratoire de Mathématiques Raphaël Salem 2 Lehigh University - Department of Mathematics

Abstract : 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.

Keywords : Random polytopes parabolic germ-grain models convex hulls of Gaussian samples Poisson point processes Burgers- equation

Autor: Pierre Calka - J. E. Yukich -

Fuente: https://hal.archives-ouvertes.fr/

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