Mixed integrals and related inequalities

TL;DR

This paper introduces a new addition operation for quasi-concave functions that enables the definition of mixed integrals, extending classical geometric inequalities like Brunn-Minkowski and Alexandrov-Fenchel to a functional framework.

Contribution

It defines a novel addition operation on quasi-concave functions that polarizes the Lebesgue integral, leading to functional analogs of mixed volumes and inequalities.

Findings

Extended Brunn-Minkowski inequality to quasi-concave functions.

Generalized Alexandrov-Fenchel inequality in the functional setting.

Proved inequalities for log-concave functions in a familiar form.

Abstract

In this paper we define an addition operation on the class of quasi-concave functions. While the new operation is similar to the well-known sup-convolution, it has the property that it polarizes the Lebesgue integral. This allows us to define mixed integrals, which are the functional analogs of the classic mixed volumes. We extend various classic inequalities, such as the Brunn-Minkowski and the Alexandrov-Fenchel inequality, to the functional setting. For general quasi-concave functions, this is done by restating those results in the language of rearrangement inequalities. Restricting ourselves to log-concave functions, we prove generalizations of the Alexandrov inequalities in a more familiar form.