%ABesau, Florian%AGusakova, Anna%AReitzner, Matthias%ASchütt, Carsten%AThäle, Christoph%AWerner, Elisabeth_M%BJournal Name: Mathematische Annalen; Journal Volume: 389; Journal Issue: 3; Related Information: CHORUS Timestamp: 2024-05-30 17:04:25
%D2023%ISpringer Science + Business Media
%JJournal Name: Mathematische Annalen; Journal Volume: 389; Journal Issue: 3; Related Information: CHORUS Timestamp: 2024-05-30 17:04:25
%K
%MOSTI ID: 10446945
%PMedium: X; Size: p. 2289-2316
%TSpherical convex hull of random points on a wedge
%XAbstract
Consider 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}\cap {H}_{1}^{+}\cap {H}_{2}^{+}$is a spherical convex subset of thed-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. Choosenindependent 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$$\log n$$$logn$. 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.

%0Journal Article