  Consider two half-spaces $H_1^+$ and $H_2^+$ in $\mathbb{R}^{d+1}$ whose
bounding hyperplanes $H_1$ and $H_2$ are orthogonal and pass through the
origin. The intersection $\mathbb{S}_{2,+}^d:=\mathbb{S}^d\cap H_1^+\cap H_2^+$
is a spherical convex subset of the $d$-dimensional unit sphere $\mathbb{S}^d$,
which contains a great subsphere of dimension $d-2$ and is called a spherical
wedge. Choose $n$ independent random points uniformly at random on
$\mathbb{S}_{2,+}^d$ and consider the expected facet number of the spherical
convex hull of these points. It is shown that, up to terms of lower order, this
expectation grows like a constant multiple of $\log n$. A similar behaviour is
obtained for the expected facet number of a homogeneous Poisson point process
on $\mathbb{S}_{2,+}^d$. The result is compared to the corresponding behaviour
of classical Euclidean random polytopes and of spherical random polytopes on a
half-sphere.
