Neighborhoods on the Grasmannian of marginals with bounded isotropic constant

Abstract

We show that for any isotropic log-concave probability measure $\mu$ on ${\mathbb{R}}^n$, for every $\epsilon >0$, every $1\le k\le \sqrt{n}$ and any $E\in G_{n,k}$ there exists $F\in G_{n,k}$ with $d(E,F)<\epsilon$ and $L_{{\pi }_F\mu }<C/\epsilon$.