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

TL;DR

This paper characterizes when certain integral functionals on convex bodies are monotonic and satisfy Brunn-Minkowski inequalities, showing these properties are linked to the functional being a mixed volume.

Contribution

It provides necessary and sufficient conditions for monotonicity and establishes the equivalence between Brunn-Minkowski inequalities and mixed volumes for a broad class of functionals.

Findings

Brunn-Minkowski inequality implies monotonicity.

Brunn-Minkowski inequality is equivalent to the functional being a mixed volume.

Necessary conditions for Brunn-Minkowski type inequalities are identified.

Abstract

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.