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. 
Andrea Colesanti, Daniel Hug, Eugenia Saor\'in-G\'omez
math.MG