Estranged facets and $k$-facets of Gaussian random point sets

TL;DR

This paper investigates the asymptotic behavior of Gaussian random polytopes in high dimensions, focusing on the expected number of facets, $k$-facets, and pairs of estranged facets as the dimension grows.

Contribution

It provides new asymptotic formulas for the expected number of facets, $k$-facets, and estranged facet pairs in high-dimensional Gaussian random polytopes.

Findings

Expected number of facets grows as $C(eta)^{d+o(d)}$ with dimension.

Extended results to the expected number of $k$-facets.

Determined the constant for the expected number of estranged facet pairs when $n=2d$.

Abstract

Gaussian random polytopes have received a lot of attention especially in the case where the dimension is fixed and the number of points goes to infinity. Our focus is on the less studied case where the dimension goes to infinity and the number of points is proportional to the dimension $d$. We study several natural quantities associated to Gaussian random polytopes in this setting. First, we show that the expected number of facets is equal to $C(\alpha)^{d+o(d)}$ where $C(\alpha)$ is some constant which depends on the constant of proportionality $\alpha$. We also extend this result to the expected number of $k$-facets. We then consider the more difficult problem of the asymptotics of the expected number of pairs of $\textit{estranged facets}$ of a Gaussian random polytope. When $n=2d$, we determined the constant $C$ so that the expected number of pairs of estranged facets is equal to $C^{d+o(d)}$.