  For a broad class of integral functionals defined on the space of
$n$-dimensional convex bodies, we establish necessary and sufficient conditions
for monotonicity, and necessary conditions for the validity of a
Brunn-Minkowski type inequality. In particular, we prove that a Brunn-Minkowski
type inequality implies monotonicity, and that a general Brunn-Minkowski type
inequality is equivalent to the functional being a mixed volume.
