Circumspheres of sets of n+1 random points in the d-dimensional Euclidean unit ball (1 <= n <= d)

摘要

In the d-dimensional Euclidean space, any set of n + 1 independent random points, uniformly distributed in the interior of a unit ball of center O, determines almost surely a circumsphere of center C and radius Omega (1 <= n <= d) and an n-flat (1 <= n <= d - 1). The orthogonal projection of O onto this flat is called O' while Delta designates the distance O'C'. The classical problem of the distance between two random points in a unit ball corresponds to n = 1. The focus is set on the family of circumspheres which are contained in this unit ball. For any d >= 2 and 1 <= n <= d-1, the joint probability density function of the distance Delta=O'C and circumradius Omega has a simple closed-form expression. The marginal probability density functions of Delta and Omega are both products of powers and a Gauss hypergeometric function. Stochastic representations of the latter random variables are described in terms of geometric means of two independent beta random variables. For n = d >= 1, Delta and Omega have a joint Dirichlet distribution with parameters (d, d(2), 1) while Delta and Omega are beta distributed. Results of Monte Carlo simulations are in very good agreement with their calculated counterparts. The tail behavior of the circumradius probability density function has been studied by Monte Carlo simulations for 2 <= n = d <= 9, where all circumspheres are this time considered, regardless of whether or not they are entirely contained in the unit ball. Published by AIP Publishing.