Abstract
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 .
2000 AMS Subject Classification:
Acknowledgments
The authors are grateful to two anonymous referees for their comments. They also wish to express their appreciation to Tiefeng Jiang and Jianqing Fan for very helpful discussion concerning Corollary 3.4.
ORCID
J. S. Brauchart http://orcid.org/0000-0003-4815-8475
A. B. Reznikov http://orcid.org/0000-0003-0603-989X
I. H. Sloan http://orcid.org/0000-0003-3769-0538
Y. G. Wang http://orcid.org/0000-0002-7450-0273
R. S.Womersley http://orcid.org/0000-0002-1271-6001
Notes
1 See also K. Fukuda, Frequently asked questions about polyhedral computation, Swiss Federal Institute of Technology, http://www.inf.ethz.ch/personal/fukudak/polyfaq/polyfaq.html, accessed August 2016.
2 The case when both d and N grow is also discussed in [CitationCai et al. 13].