On a geometric combination of functions related to Pr\'ekopa-Leindler inequality

TL;DR

This paper introduces a new geometric combination operation for functions based on mass transportation, leading to a novel Prékopa-Leindler type inequality and applications to convex geometry.

Contribution

It proposes a new geometric combination of functions using inverse distribution functions, providing alternative proofs for key inequalities in convex geometry.

Findings

The Lebesgue integral of the geometric combination equals the geometric mean of the integrals.

Derived a new functional inequality of Prékopa-Leindler type.

Provided an alternative proof of the log-Brunn-Minkowski inequality for specific convex bodies.

Abstract

We introduce a new operation between nonnegative integrable functions on $\mathbb{R} ^n$, that we call geometric combination; it is obtained via a mass transportation approach, playing with inverse distribution functions. The main feature of this operation is that the Lebesgue integral of the geometric combination equals the geometric mean of the two separate integrals; as a natural consequence, we derive a new functional inequality of Pr\'ekopa-Leindler type. When applied to the characteristic functions of two measurable sets, their geometric combination provides a set whose volume equals the geometric mean of the two separate volumes. In the framework of convex bodies, by comparing the geometric combination with the $0$-sum, we get an alternative proof of the log-Brunn-Minkowski inequality for unconditional convex bodies and for convex bodies with $n$ symmetries.