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@article{EJP2150,
	author = {Radosław Adamczak and Olivier Guédon and Rafał Latała and Alexander Litvak and Krzysztof Oleszkiewicz and Alain Pajor and Nicole Tomczak-Jaegermann},
	title = {Moment estimates for convex measures},
	journal = {Electron. J. Probab.},
	fjournal = {Electronic Journal of Probability},
	volume = {17},
	year = {2012},
	keywords = {convex measures,  \$\kappa\$-concave measure, tail inequalities, small ball probability estimate.},
	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 $$(\mathbb{E} |X|^{-p} )^{-1/{p}} \geq  c(\varepsilon) \left( \mathbb{E} |X| - C \sigma_{p}(X)\right) . $$
},
	pages = {no. 101, 1-19},
	issn = {1083-6489},
	doi = {10.1214/EJP.v17-2150},    
        url = {http://ejp.ejpecp.org/article/view/2150}}