On nearly radial marginals of high-dimensional probability measures

Abstract

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