  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
inequality for the mixed $f$-divergence are proved.
