Random circles in the d-dimensional unit ball

Abstract

This note gives the solution of the following problem concerning geometric probabilities. What is the probabilityp(Bd; 2) that the circumference determined by three pointsP, P1andP2chosen independently and uniformly at random in the interior of ad-dimensional unit ballBdin Euclidean spaceEd(d≧ 2) is entirely contained inBd? From our result we conclude thatp(Bd; 2) →π /(3√3) asd→∞.