Monotonicity and concavity of integral functionals involving area measures of convex bodies

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.