A short proof of Paouris' inequality
  We give a short proof of a result of G. Paouris on the tail behaviour of the Euclidean norm $|X|$ of an isotropic log-concave random vector $X\in\R^n$, stating that for every $t\geq 1$, $P(|X|\geq ct\sqrt n)\leq \exp(-t\sqrt n)$. More precisely we show that for any log-concave random vector $X$ and any $p\geq 1$, $(E|X|^p)^{1/p}\sim E |X|+\sup_{z\in S^{n-1}}(E |< z,X>|^p)^{1/p}$. 
Rados{\l}aw Adamczak, Rafa{\l} Lata{\l}a, Alexander E. Litvak,   Krzysztof Oleszkiewicz, Alain Pajor, Nicole Tomczak-Jaegermann
math.PR math.MG