Spherical convex hull of random points on a wedge

Abstract

AbstractConsider two half-spaces $$H_1^+$$ H 1 + and $$H_2^+$$ H 2 + in $${\mathbb {R}}^{d+1}$$ R d + 1 whose bounding hyperplanes $$H_1$$ H 1 and $$H_2$$ H 2 are orthogonal and pass through the origin. The intersection $${\mathbb {S}}_{2,+}^d:={\mathbb {S}}^d\cap H_1^+\cap H_2^+$$ S 2 , + d : = S d ∩ H 1 + ∩ H 2 + is a spherical convex subset of the d-dimensional unit sphere $${\mathbb {S}}^d$$ S d , which contains a great subsphere of dimension $$d-2$$ d - 2 and is called a spherical wedge. Choose n independent random points uniformly at random on $${\mathbb {S}}_{2,+}^d$$ 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 $$\log n$$ log n . A similar behaviour is obtained for the expected facet number of a homogeneous Poisson point process on $${\mathbb {S}}_{2,+}^d$$ 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.