# Euclidean Forward-Reverse Brascamp-Lieb Inequalities: Finiteness, Structure and Extremals

## TL;DR

This paper extends the Brascamp-Lieb inequalities by providing a new proof of Gaussian extremality, developing a duality principle for constants, and generalizing key results on finiteness, structure, and extremizers to the forward-reverse setting.

## Contribution

It introduces a new proof for Gaussian extremizers, a duality principle for best constants, and generalizes finiteness and structure results to the forward-reverse Brascamp-Lieb inequalities.

## Key findings

- Centered Gaussian functions saturate the inequalities.
- A duality principle for the best constants is established.
- Main results on finiteness, structure, and extremizers are generalized.

## Abstract

A new proof is given for the fact that centered gaussian functions saturate the Euclidean forward-reverse Brascamp-Lieb inequalities, extending the Brascamp-Lieb and Barthe theorems. A duality principle for best constants is also developed, which generalizes the fact that the best constants in the Brascamp-Lieb and Barthe inequalities are equal. Finally, as the title hints, the main results concerning finiteness, structure and gaussian-extremizability for the Brascamp-Lieb inequality due to Bennett, Carbery, Christ and Tao are generalized to the setting of the forward-reverse Brascamp-Lieb inequality.