Transport-majorization to analytic and geometric inequalities

TL;DR

This paper introduces a novel transport-majorization method to derive geometric and Fourier inequalities, demonstrating that strongly log-concave densities are majorized by Gaussian densities, which maximize certain entropy measures.

Contribution

The paper presents a new transport-majorization technique for establishing convex order relations and applies it to derive inequalities and entropy maximization results.

Findings

Strongly log-concave densities majorize Gaussian densities.

Gaussian densities maximize Rényi and Tsallis entropies among strongly log-concave densities.

Elementary derivations of Fourier and geometric inequalities using transport-majorization.

Abstract

We introduce a transport-majorization argument that establishes a majorization in the convex order between two densities, based on control of the gradient of a transportation map between them. As applications, we give elementary derivations of some delicate Fourier analytic inequalities, which in turn yield geometric "slicing-inequalities" in both continuous and discrete settings. As a further consequence of our investigation we prove that any strongly log-concave probability density majorizes the Gaussian density and thus the Gaussian density maximizes the R\'enyi and Tsallis entropies of all orders among all strongly log-concave densities.