Portal de Periódicos da CAPES

Geometrical Probability and Random Points on a Hypersphere

1967; Institute of Mathematical Statistics; Volume: 38; Issue: 1 Linguagem: Inglês

10.1214/aoms/1177699073

2168-8990

Thomas M. Cover, Bradley Efron,

Computational Geometry and Mesh Generation

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 $E^d$.