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In the d dimensional Euclidean space, any set of n+1 independent randompoints, 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 npositive 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 oncircumspheres which are contained in this unit ball. For any d larger than 1and any n positive and less than d, the joint probability density function ofthe distance D = O'C and of R has a simple closed form expression. Theirmarginal probability density functions are both products of powers and of aGauss hypergeometric function. Stochastic representations of D and R areexpressed as geometric means of two independent beta random variables. For nequal to d, positive, D and R have a Dirichlet distribution while D and R areeach beta distributed. Results of Monte-Carlo simulations are in very goodagreement with their calculated counterparts

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