On nearly radial marginals of high-dimensional probability measures

Abstract

Suppose that μ is an absolutely continuous probability measure on ℝn, for large n. Then μ has low-dimensional marginals that are approximately spherically-symmetric. More precisely, if n ≥ (C/epse;)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. © 2010 European Mathematical Society.