Curvature functionals on convex bodies

Abstract

We investigate the weighted Lp affine surface areas which appear in the recently established Lp Steiner formula of the Lp Brunn Minkowski theory. We show that they are valuations on the set of convex bodies and prove isoperimetric inequalities for them. 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 Rényi-divergence.