Circumspheres of sets of n+1 random points in the d-dimensional Euclidean unit ball (0<n<d+1)

TL;DR

This paper derives explicit probability distributions for the circumsphere center distance and radius of n+1 random points inside a d-dimensional unit ball, revealing their stochastic structure and providing simulation validation.

Contribution

It provides closed-form expressions and stochastic representations for the distributions of circumsphere parameters for random points in a unit ball, extending geometric probability theory.

Findings

Explicit joint probability density functions for D and R.

Stochastic representations as geometric means of beta variables.

Monte Carlo simulations confirm theoretical results.

Abstract

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 R, with n positive and less than d+1, and a n flat when n is positive and less than d. The projection of O on the n flat is named O'. The focus is set on circumspheres which are contained in this unit ball. For any d larger than 1 and any n positive and less than d, the joint probability density function of the distance D = O'C and of R has a simple closed form expression. Their marginal probability density functions are both products of powers and of a Gauss hypergeometric function. Stochastic representations of D and R are expressed as geometric means of two independent beta random variables. For n equal to d, positive, D and R have a Dirichlet distribution while D and R are each beta distributed. Results of Monte-Carlo simulations are in very good agreement with their calculated counterparts