Facets of High-Dimensional Gaussian Polytopes

Abstract

AbstractWe study the number of facets of the convex hull of n independent standard Gaussian points in $${\mathbb {R}}^d$$ 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 → 0 . We also obtain the asymptotic value when d is close to n.