Functional Brunn-Minkowski Inequalities Induced by Polarity

TL;DR

This paper introduces a new family of inequalities based on a geometric convolution derived from the polarity transform, extending classical inequalities like Brunn-Minkowski and connecting to convexity theorems.

Contribution

It develops a novel geometric inf-convolution linked to the polarity transform, leading to new inequalities that generalize and extend classical convex geometric results.

Findings

Proved a new inequality relating geometric convolution integrals to harmonic averages.

Derived inequalities that imply the classical Brunn-Minkowski inequality.

Established a new variant of Busemann's convexity theorem for 1-convex hulls and log-concave densities.

Abstract

We prove a new family of inequalities, which compare the integral of a geometric convolution of non-negative functions with the integrals of the original functions. For classical inf-convolution, this type of inequality is called the Pr\'{e}kopa-Leindler inequality, which, restricted to indicators of convex bodies, gives the classical Brunn-Minkowski inequality. The convolution we consider is a different one, which arises from the study of the polarity transform for functions. While inf-convolution arises as the pull back of usual addition of convex functions under the Legendre transform, our geometric inf-convolution arises as the pull back of the second order reversing transform on geometric convex functions (called either polarity transform or $A$-transform). These are, up to linear terms, the only order reversing isomorphisms on the class of geometric convex functions. We prove that the integral of this new geometric convolution of two functions is bounded from below by the harmonic average of the individual integrals. Our inequality implies the Brunn-Minkowski inequality, as well as some other, new, inequalities for volumes of bodies. Our inequalities are intimately connected with Busemann's convexity theorem, a new variant of which we prove for 1-convex hulls and log-concave densities.