On nearly radial marginals of high-dimensional probability measures

Abstract

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.