Geometrical Probability and Random Points on a Hypersphere

This paper is concerned with the properties of convex cones and their dual cones generated by points randomly distributed on the surface of a $d$-sphere. For radially symmetric distributions on the points, the expected number of $k$-faces and natural measure of the set of $k$-faces will be found. The expected number of vertices, or extreme points, of convex hulls of random points in $E^2$ and $E^3$ has been investigated by Renyi and Sulanke [4] and Efron [2]. In general these results depend critically on the distribution of the points. However, for points on a sphere, the situation is much simpler. Except for a requirement of radial symmetry of the distribution on the points, the properties developed in this paper will be distribution-free. (This lack of dependence on the underlying distribution suggests certain simple nonparametric tests for radial symmetry--we shall not pursue this matter here, however.) Our approach is combinatorial and geometric, involving the systematic description of the partitioning of $E^d$ by $N$ hyperplanes through the origin. After a series of theorems counting the number of faces of cones and their duals, we are led to Theorem 5 and its probabilistic counterpart Theorem 2', the primary result of this paper, in which the expected solid angle is found of the convex cone spanned by $N$ random vectors in $E^d$.