Moment comparisons, Sudakov inequalities and entropy of centroid bodies

Abstract

Let $X$ be an isotropic log-concave random vector in ${\mathbb{R}}^n$ and let $G$ be standard Gaussian. Starting from the first moment comparison for gauges, we derive corresponding estimates at arbitrary moment orders. For every gauge $\varphi$ and every $q⩾1$, $$\Vert \varphi (G){\Vert }_q⩽C\sqrt{\ln (en)+q}\Vert \varphi (X){\Vert }_q,\Vert \varphi (X){\Vert }_q⩽C\left(\sqrt{\ln (en)}+\psi (X)\sqrt{q}\right)\Vert \varphi (G){\Vert }_q.$$ Applied to support functions, this gives the sharp worst case order $C\sqrt{\ln (en)}$ for the $L_2$-Sudakov constant and quantitative $L_p$-Sudakov estimates. Combining, at each level of Latala's dyadic chain, the strongest of the three quantitative $L_p$-Sudakov estimates used here yields $${\left(\mathbb{E}\Vert Y{\Vert }^p\right)}^{1/p}⩽C\left(n^{1/4}\sqrt{\ln (en)}\ln (e+\ln (en))\mathbb{E}\Vert X\Vert +{\sigma }_p(Y)\right)$$ whenever the weak moments of $Y$ are dominated by those of $X$. In a second direction, we study the generalized dual Sudakov problem for the self generated metrics associated with $Z_r(X{)}^{\circ }$ and prove dimension free packing estimates for $Z_p(X)$. We also obtain mean norm estimates for centroid bodies, factorization through arbitrary symmetric convex bodies and an affine dimensional refinement.