Facets of high-dimensional Gaussian polytopes

TL;DR

This paper investigates the asymptotic behavior of the number of facets of convex hulls formed by high-dimensional Gaussian points, providing explicit formulas for various growth regimes of dimension relative to sample size.

Contribution

It derives explicit asymptotic formulas for the expected number of facets of Gaussian polytopes as dimension grows with sample size, covering different regimes.

Findings

Explicit asymptotic formula for d/n → 0

Asymptotic value when d is close to n

Insights into high-dimensional Gaussian convex hulls

Abstract

We study the number of facets of the convex hull of n independent standard Gaussian points in d-dimensional Euclidean space. 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 tends to zero. We also obtain the asymptotic value when d is close to n.