Quantitative stability for the Brascamp-Lieb inequality and moment measures

TL;DR

This paper establishes sharp quantitative stability results for the Brascamp--Lieb inequality and moment measures, emphasizing the independence of stability constants from the convex functions involved, with potential broad applications.

Contribution

It introduces a novel stability constant for the Brascamp--Lieb inequality independent of the convex function, and provides uniform stability results for moment measures.

Findings

Sharp stability version for Brascamp--Lieb inequality

Stability results for moment measures are uniform

Stability constants are independent of the convex function

Abstract

By employing the recently obtained sharp stability versions of the Pr\'ekopa--Leindler inequality, we are able to obtain a sharp quantitative stability version for the Brascamp--Lieb inequality, as well as several different results on the stability of moment measures. As main features of our results, we highlight the independence of the stability constant for the Brascamp--Lieb inequality on the convex function considered, a completely novel feature. In the realm of moment measures, we highlight in the same vein that the stability results obtained are uniform, which we expect to be particularly valuable not only from a purely mathematical point of view, but also for applications.