\alpha-concave functions and a functional extension of mixed volumes

TL;DR

This paper introduces mixed integrals as functional analogs of mixed volumes for \

Contribution

It constructs a natural addition operation for \

Findings

Defined mixed integrals as polarization of the integral for quasi-concave functions.

Extended classical inequalities to the functional setting, including a generalization of Alexandrov inequalities.

Established a framework for \

Abstract

Mixed volumes, which are the polarization of volume with respect to the Minkowski addition, are fundamental objects in convexity. In this note we announce the construction of mixed integrals, which are functional analogs of mixed volumes. We build a natural addition operation + on the class of quasi-concave functions, such that every class of \alpha-concave functions is closed under +. We then define the mixed integrals, which are the polarization of the integral with respect to +. We proceed to discuss the extension of various classic inequalities 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 \alpha-concave functions, we state a generalization of the Alexandrov inequalities in their more familiar form.