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 .

[1]  M. Nakagami.  The m-Distribution—A General Formula of Intensity Distribution of Rapid Fading , 1960 .

[2]  J. G. Wendel.  A Problem in Geometric Probability. , 1962 .

[3]  R. E. Miles.  ISOTROPIC RANDOM SIMPLICES , 1971 .

[4]  L. Shepp.  Covering the circle with random ARCS , 1972 .

[5]  R. Tichy,et al.  Stochastical approximation of convex bodies , 1985 .

[6]  H. Maehara.  A threshold for the size of random caps to cover a sphere , 1988 .

[7]  J. Møller.  Random tessellations in ℝ d , 1989, Advances in Applied Probability.

[8]  Josef S. Müller.  Approximation of a ball by random polytopes , 1990 .

[9]  E. Saff,et al.  Asymptotics for minimal discrete energy on the sphere , 1995 .

[10]  M. Reitzner.  Random points on the boundary of smooth convex bodies , 2002 .

[11]  C. Schütt,et al.  Polytopes with Vertices Chosen Randomly from the Boundary of a Convex Body , 2003 .

[12]  J. R,et al.  Estimator Comparison of the Nakagami-m Parameter and Its Application in Echocardiography , 2004 .

[13]  Ian H. Sloan,et al.  Extremal Systems of Points and Numerical Integration on the Sphere , 2004, Adv. Comput. Math..

[14]  H. N. Nagaraja,et al.  Order Statistics, Third Edition , 2005, Wiley Series in Probability and Statistics.

[15]  Volker Schönefeld.  Spherical Harmonics , 2019, An Introduction to Radio Astronomy.

[16]  Ronald F. Boisvert,et al.  NIST Handbook of Mathematical Functions , 2010 .

[17]  F. Cucker,et al.  Coverage processes on spheres and condition numbers for linear programming , 2007, 0712.2816.

[18]  Herbert A. David,et al.  Order Statistics , 2011, International Encyclopedia of Statistical Science.

[19]  D. Hug,et al.  Intrinsic volumes of random polytopes with vertices on the boundary of a convex body , 2011, 1103.4978.

[20]  P. Sarnak,et al.  Local Statistics of Lattice Points on the Sphere , 2012, 1204.0134.

[21]  E. Saff,et al.  The next-order term for optimal Riesz and logarithmic energy asymptotics on the sphere , 2012, 1202.4037.

[22]  Jianqing Fan,et al.  Distributions of angles in random packing on spheres , 2013, J. Mach. Learn. Res..

[23]  P. Calka,et al.  Extreme values for characteristic radii of a Poisson-Voronoi Tessellation , 2013, 1304.0170.

[24]  Gelasio Salazar,et al.  Large convex holes in random point sets , 2013, Comput. Geom..

[25]  E. Saff,et al.  The Covering Radius of Randomly Distributed Points on a Manifold , 2015, 1504.03029.

[26]  Gelasio Salazar,et al.  Large Area Convex Holes in Random Point Sets , 2015, SIAM J. Discret. Math..