Limit theorems for uniform distributions on spheres in high-dimensional euclidean spaces

If X = ( X 1 , · ··, X n ) has uniform distribution on the sphere or ball in ℝ with radius a, then the joint distribution of , ···, k, converges in total variation to the standard normal distribution on ℝ. Similar results hold for the inner products of independent n -vectors. Applications to geometric probability are given.