Expected $f$-vector of the Poisson Zero Polytope and Random Convex Hulls in the Half-Sphere

We prove an explicit combinatorial formula for the expected number of faces of the zero polytope of the homogeneous and isotropic Poisson hyperplane tessellation in ${\mathbb{R}}^d$. The expected $f$-vector is expressed through the coefficients of the polynomial $$(1+(d-1{)}^2{x}^2)(1+(d-3{)}^2{x}^2)(1+(d-5{)}^2{x}^2)\dots \mathrm{.}$$ Also, we compute explicitly the expected $f$-vector and the expected volume of the spherical convex hull of $n$ random points sampled uniformly and independently from the $d$-dimensional half-sphere. In the case when $n=d+2$, we compute the probability that this spherical convex hull is a spherical simplex, thus solving an analogue of the Sylvester four-point problem on the half-sphere.