Moment estimates for convex measures

I will present estimates of $p$-th moments of the Euclidean norm for a certain class of measures, satisfying a Brunn-Minkowski type inequality, in terms of the first moment and weak $p$-th moments and their applications to tail estimates (positive $p$) and small ball inequalities (negative $p$). The talk will be based on joint work with O. Guédon, R. Latała, A. Litvak, K. Oleszkiewicz, A. Pajor and N. Tomczak-Jaegermann.