A recent simple proof of the Paouris bound on deviations of the Euclidean norm of the logarithmically concave random vector will be presented. The main tool used in it is a classical min-max Slepian-type inequality of Y. Gordon. Joint work with Radoslaw Adamczak, Rafal Latala, Alexander Litvak, Alain Pajor, and Nicole Tomczak-Jaegermann.