The geometry of Euclidean convolution inequalities and entropy

TL;DR

This paper demonstrates that key convolution inequalities in Harmonic Analysis and Information Theory can be understood through a geometric perspective in -dimensional space, leading to direct derivations of entropic inequalities.

Contribution

It introduces a geometric approach to derive and understand convolution and entropic inequalities, connecting harmonic analysis, information theory, and geometry.

Findings

Derivation of entropic inequalities from geometric analysis

Reduction of convolution inequalities to geometric problems in 

Establishment of duality between entropic and convolution inequalities

Abstract

The goal of this note is to show that some convolution type inequalities from Harmonic Analysis and Information Theory, such as Young's convolution inequality (with sharp constant), Nelson's hypercontractivity of the Hermite semi-group or Shannon's inequality, can be reduced to a simple geometric study of frames of $\R^2$. We shall derive directly entropic inequalities, which were recently proved to be dual to the Brascamp-Lieb convolution type inequalities.