Moment estimates for convex measures

Abstract

Let $p\geq 1$, $\varepsilon >0$, $r\geq (1+\varepsilon) p$, and $X$ be a $(-1/r)$-concave random vector in $\mathbb{R}^n$ with Euclidean norm $|X|$. We prove that $$(\mathbb{E} |X|^{p})^{1/{p}}\leq c \left( C(\varepsilon) \mathbb{E} |X|+\sigma_{p}(X)\right), $$ where $$\sigma_{p}(X) = \sup_{|z|\leq 1}(\mathbb{E} |\langle z,X\rangle|^{p})^{1/p}, $$ $C(\varepsilon)$ depends only on $\varepsilon$ and $c$ is a universal constant. Moreover, if in addition $X$ is centered then $$(\mathbb{E} |X|^{-p} )^{-1/{p}} \geq c(\varepsilon) \left( \mathbb{E} |X| - C \sigma_{p}(X)\right) . $$