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

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 .

Keywords:

• spherical random points
• covering radius
• moments of hole radii
• point separation
• random polytopes

2000 AMS Subject Classification:

• Primary 52C17
• 52A22
• Secondary 60D05

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 [Cai et al. 13].

Additional information

Funding

This research was supported under the Australian Research Council’s Discovery Projects funding scheme (project number DP120101816). The research of J. S. Brauchart was also supported by the Austrian Science Fund FWF projects F5510 (part of the Special Research Program (SFB) “Quasi-Monte Carlo Methods: Theory and Applications”). The research of A. B. Reznikov and E. B. Saff was also supported by the U.S. National Science Foundation grants DMS-1412428 and DMS-1516400. The authors also acknowledge the support of the Erwin Schrödinger Institute in Vienna, where part of the work was carried out.