  In this paper, the Orlicz addition of measures is proposed and an
interpretation of the $f$-divergence is provided based on a linear Orlicz
addition of two measures. Fundamental inequalities, such as, a dual functional
Orlicz-Brunn-Minkowski inequality, are established. We also investigate an
optimization problem for the $f$-divergence and establish functional affine
isoperimetric inequalities for the dual functional Orlicz affine and geominimal
surface areas of measures.
