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Title: Spherical convex hull of random points on a wedge
Abstract

Consider two half-spaces$$H_1^+$$H1+and$$H_2^+$$H2+in$${\mathbb {R}}^{d+1}$$Rd+1whose bounding hyperplanes$$H_1$$H1and$$H_2$$H2are orthogonal and pass through the origin. The intersection$${\mathbb {S}}_{2,+}^d:={\mathbb {S}}^d\cap H_1^+\cap H_2^+$$S2,+d:=SdH1+H2+is a spherical convex subset of thed-dimensional unit sphere$${\mathbb {S}}^d$$Sd, which contains a great subsphere of dimension$$d-2$$d-2and is called a spherical wedge. Choosenindependent random points uniformly at random on$${\mathbb {S}}_{2,+}^d$$S2,+dand 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$${\mathbb {S}}_{2,+}^d$$S2,+d. 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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NSF-PAR ID:
10446945
Author(s) / Creator(s):
; ; ; ; ;
Publisher / Repository:
Springer Science + Business Media
Date Published:
Journal Name:
Mathematische Annalen
ISSN:
0025-5831
Format(s):
Medium: X
Sponsoring Org:
National Science Foundation
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