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 $\mathrm{\mid }X\mathrm{\mid }$ of an isotropic log-concave random vector $X\in {\mathbb{R}}^n$, stating that for every $t\ge 1$, $P(\mathrm{\mid }X\mathrm{\mid }\ge ct\sqrt{n})\le \exp (-t\sqrt{n})$. More precisely we show that for any log-concave random vector $X$ and any $p\ge 1$, $(E\mathrm{\mid }X{\mathrm{\mid }}^p{)}^{1\mathrm{/}p}\sim E\mathrm{\mid }X\mathrm{\mid }+{\sup }_{z\in S^{n-1}}(E\mathrm{\mid }<z,X>{\mathrm{\mid }}^p{)}^{1\mathrm{/}p}$.