  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
