Geometry of the $L_q$-centroid bodies of an isotropic log-concave measure

Abstract

We study some geometric properties of the $L_q$-centroid bodies $Z_q(\mu )$ of an isotropic log-concave measure $\mu$ on ${\mathbb{R}}^n$. For any $2\ls q\ls\sqrt{n}$ and for $\epsilon \in ({\epsilon }_0(q,n),1)$ we determine the inradius of a random $(1-\epsilon )n$-dimensional projection of $Z_q(\mu )$ up to a constant depending polynomially on $\epsilon$. Using this fact we obtain estimates for the covering numbers $N(\sqrt{q}{B}_2^n,t{Z}_q(\mu ))$, $t\gr 1$, thus showing that $Z_q(\mu )$ is a $\beta$-regular convex body. As a consequence, we also get an upper bound for $M(Z_q(\mu ))$.