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Article Dans Une Revue Journal of Mathematical Physics Année : 2017

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

G Le Caër
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Résumé

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 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.
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Dates et versions

hal-01517928 , version 1 (03-05-2017)

Identifiants

Citer

G Le Caër. Circumspheres of sets of n+1 random points in the d-dimensional Euclidean unit ball (1≤n≤d). Journal of Mathematical Physics, 2017, 58 (5), pp.053301. ⟨10.1063/1.4982640⟩. ⟨hal-01517928⟩
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