$M$-estimates for isotropic convex bodies and their $L_q$-centroid bodies

Abstract

Let $K$ be a centrally-symmetric convex body in $\mathbb{R}^n$ and let $\|\cdot\|$ be its induced norm on ${\mathbb R}^n$. We show that if $K \supseteq r B_2^n$ then: \[ \sqrt{n} M(K) \leqslant C \sum_{k=1}^{n} \frac{1}{\sqrt{k}} \min\left(\frac{1}{r} , \frac{n}{k} \log\Big(e + \frac{n}{k}\Big) \frac{1}{v_{k}^{-}(K)}\right) . \] where $M(K)=\int_{S^{n-1}} \|x\|\, d\sigma(x)$ is the mean-norm, $C>0$ is a universal constant, and $v^{-}_k(K)$ denotes the minimal volume-radius of a $k$-dimensional orthogonal projection of $K$. We apply this result to the study of the mean-norm of an isotropic convex body $K$ in ${\mathbb R}^n$ and its $L_q$-centroid bodies. In particular, we show that if $K$ has isotropic constant $L_K$ then: \[ M(K) \leqslant \frac{C\log^{2/5}(e+ n)}{\sqrt[10]{n}L_K} . \]