  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 $C^{d+o(d)}$.
