On the mixed $f$-divergence for multiple pairs of measures

In this paper, the concept of the classical f -divergence (for a pair of measures) is extended to the mixed f divergence (for multiple pairs of measures). The mixed f -divergence provides a way to measure the difference between multiple pairs of (probability) measures. Properties for the mixed f -divergence are established, such as permutation invariance and symmetry in distributions. An Alexandrov-Fenchel type inequality and an isoperimetric type inequality for the mixed f -divergence will be proved and applications in the theory of convex bodies are given. Index Terms Alexandrov-Fenchel inequality, f -dissimilarity, f -divergence, isoperimetric inequality.