On nearly radial marginals of high-dimensional probability measures

Suppose that μ is an absolutely continuous probability measure on ℝ^n , for large n . Then μ has low-dimensional marginals that are approximately spherically-symmetric. More precisely, if n ≥ (C/ε)^{Cd} , then there exist d -dimensional marginals of μ that are ε -far from being spherically-symmetric, in an appropriate sense. Here C > 0 is a universal constant.