Face numbers of high-dimensional Poisson zero cells

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 $d\to\infty$.