Random Point Sets on the Sphere-Hole Radii, Covering, and Separation

Geometric properties of N random points distributed independently and uniformly on the unit sphere with respect to surface area measure are obtained and several related conjectures are posed. In particular, we derive asymptotics (as N → ∞) for the expected moments of the radii of spherical caps asso...

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Bibliographic Details
Published in:Experimental mathematics 2018-01, Vol.27 (1), p.62-81
Main Authors: Brauchart, J. S., Reznikov, A. B., Saff, E. B., Sloan, I. H., Wang, Y. G., Womersley, R. S.
Format: Article
Language:English
Subjects:
covering radius
moments of hole radii
point separation
Primary 52C17
random polytopes
Secondary 60D05
spherical random points
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Summary:Geometric properties of N random points distributed independently and uniformly on the unit sphere with respect to surface area measure are obtained and several related conjectures are posed. In particular, we derive asymptotics (as N → ∞) for the expected moments of the radii of spherical caps associated with the facets of the convex hull of N random points on . We provide conjectures for the asymptotic distribution of the scaled radii of these spherical caps and the expected value of the largest of these radii (the covering radius). Numerical evidence is included to support these conjectures. Furthermore, utilizing the extreme law for pairwise angles of Cai et al., we derive precise asymptotics for the expected separation of random points on .
ISSN:1058-6458
1944-950X
DOI:10.1080/10586458.2016.1226209