'a random vector follows uniform distribution on the unit sphere', I wonder is based on the before-rotation vectors are uniformly distributed(since the random rotation matrix is shared by all vectors!). Otherwise, assume all vectors are concentrated in 1 dimension and stay very close, after applying a random rotation matrix on them, they still possibly stay in 1 dimension and the theorem can not hold.
'a random vector follows uniform distribution on the unit sphere', I wonder is based on the before-rotation vectors are uniformly distributed(since the random rotation matrix is shared by all vectors!). Otherwise, assume all vectors are concentrated in 1 dimension and stay very close, after applying a random rotation matrix on them, they still possibly stay in 1 dimension and the theorem can not hold.