We show isoperimetric inequalities for weighted L_p affine surface areas which appear in the recently established L_p Steiner formula of the L_p Brunn Minkowski theory. We show that they are related to f-divergences of the cone measures of the convex body and its polar, namely the Kullback-Leibler divergence and the Renyi-divergence. Based on joint work with Kateryna Tatarko.