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

International audienceIn 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 of radius Omega (1≤n≤d) and a 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 any1≤n≤d-1, the joint probability density function of the distance O’C and of the circumradius has a simple closed-form expression. The marginal probability density functions of Delta and of Omega are both products of powers and of 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