  Let $\mathcal Z_d$ be the zero cell of a $d$-dimensional, isotropic and
stationary Poisson hyperplane tessellation. We study the asymptotic behavior of
the expected number of $k$-dimensional faces of $\mathcal Z_d$, as
$d\to\infty$. For example, we show that the expected number of hyperfaces of
$\mathcal Z_d$ is asymptotically equivalent to $\sqrt{2\pi/3}\, d^{3/2}$, as
$d\to\infty$. We also prove that the expected solid angle of a random cone
spanned by $d$ random vectors that are independent and uniformly distributed on
the unit upper half-sphere in $\mathbb R^{d}$ is asymptotic to $\sqrt 3
\pi^{-d}$, as $d\to\infty$.
