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We prove an explicit combinatorial formula for the expected number of facesof the zero polytope of the homogeneous and isotropic Poisson hyperplanetessellation in $\mathbb R^d$. The expected $f$-vector is expressed through thecoefficients of the polynomial $$ (1+ (d-1)^2x^2) (1+(d-3)^2 x^2) (1+(d-5)^2x^2) \ldots. $$ Also, we compute explicitly the expected $f$-vector and theexpected volume of the spherical convex hull of $n$ random points sampleduniformly and independently from the $d$-dimensional half-sphere. In the casewhen $n=d+2$, we compute the probability that this spherical convex hull is aspherical simplex, thus solving an analogue of the Sylvester four-point problemon the half-sphere.

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