Mixed f-divergence and inequalities for log-concave functions

Abstract

Mixed |$f$|-divergences, a concept from information theory and statistics, measure the difference between multiple pairs of distributions. We introduce them for log-concave functions and establish some of their properties. Among them are affine invariant vector entropy inequalities, like new Alexandrov–Fenchel-type inequalities and an affine isoperimetric inequality for the vector form of the Kullback Leibler divergence for log-concave functions.

Special cases of |$f$|-divergences are mixed |$L_\lambda$|-affine surface areas for log-concave functions. For those, we establish various affine isoperimetric inequalities as well as a vector Blaschke Santaló-type inequality.