Estranged facets and <i>k</i> -facets of Gaussian random point sets

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 with 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 estranged facets of a Gaussian random polytope. When the number of points is 2 d , we determine the constant C such that the expected number of pairs of estranged facets is equal to $C^{d+o(d)}$ .