Facets of High-Dimensional Gaussian Polytopes

We study the number of facets of the convex hull of n independent standard Gaussian points in \({\mathbb {R}}^d\) ${\mathbb{R}}^d$ . In particular, we are interested in the expected number of facets when the dimension is allowed to grow with the sample size. We establish an explicit asymptotic formula that is valid whenever \(d/n\rightarrow 0\) $d/n\to 0$ . We also obtain the asymptotic value when d is close to n.