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