Consider two halfspaces
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Abstract and$$H_1^+$$ ${H}_{1}^{+}$ in$$H_2^+$$ ${H}_{2}^{+}$ whose bounding hyperplanes$${\mathbb {R}}^{d+1}$$ ${R}^{d+1}$ and$$H_1$$ ${H}_{1}$ are orthogonal and pass through the origin. The intersection$$H_2$$ ${H}_{2}$ is a spherical convex subset of the$${\mathbb {S}}_{2,+}^d:={\mathbb {S}}^d\cap H_1^+\cap H_2^+$$ ${S}_{2,+}^{d}:={S}^{d}\cap {H}_{1}^{+}\cap {H}_{2}^{+}$d dimensional unit sphere , which contains a great subsphere of dimension$${\mathbb {S}}^d$$ ${S}^{d}$ and is called a spherical wedge. Choose$$d2$$ $d2$n independent random points uniformly at random on 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$${\mathbb {S}}_{2,+}^d$$ ${S}_{2,+}^{d}$ . A similar behaviour is obtained for the expected facet number of a homogeneous Poisson point process on$$\log n$$ $logn$ . The result is compared to the corresponding behaviour of classical Euclidean random polytopes and of spherical random polytopes on a halfsphere.$${\mathbb {S}}_{2,+}^d$$ ${S}_{2,+}^{d}$ 
Reitzner, Matthias ; Schütt, Carsten ; Werner, Elisabeth M. ( , Discrete & Computational Geometry)
Abstract The convex hull of
N independent random points chosen on the boundary of a simple polytope in is investigated. Asymptotic formulas for the expected number of vertices and facets, and for the expectation of the volume difference are derived. This is one of the first investigations leading to rigorous results for random polytopes which are neither simple nor simplicial. The results contrast existing results when points are chosen in the interior of a convex set.$$ {\mathbb {R}}^n$$ ${R}^{n}$