Spherical convex hull of random points on a wedge

Consider two half-spaces H + 1 and H + 2 in R d + 1 whose bounding hyperplanes H1 and H2 are orthogonal and pass through the origin. The intersection S d 2 ,+ := S d ∩ H + 1 ∩ H + 2 is a spherical convex subset of the d-dimensional unit sphere 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 S d 2 ,+ 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 S d 2 ,+ . The result is compared to the corresponding behaviour of classical Euclidean random polytopes and of spherical random polytopes on a half-sphere.

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