Quantitative stability for the Brascamp-Lieb inequality and moment measures
João Miguel Machado, João P. G. Ramos
TL;DR
The paper establishes sharp, dimension-dependent stability results for the Brascamp–Lieb variance inequality and for moment measures by leveraging sharp Prékopa–Leindler stability and optimal transport tools. It proves a novel L1-stability bound for the Brascamp–Lieb inequality with a constant independent of the convex potential, and derives uniform stability results for moment measures, including backbone estimates and convergence rates under regularization. The results extend to both compact domains and the full space ℝ^d under natural geometric conditions (Θ lower bounds or curvature bounds), yielding explicit rates in Wasserstein distances that are suitable for numerical methods and sampling applications. Together, these findings strengthen the link between functional-analytic inequalities and OT-based representations of measures, with practical implications for moment-measure representations and their computational realization.
Abstract
By employing the recently obtained sharp stability versions of the Prékopa--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.