Quantitative stability for the Brascamp-Lieb inequality and moment measures

We develop a quantitative stability theory for moment measures based on a new sharp uniform stability principle for the Brascamp-Lieb variance inequality in terms of the $L^1$-distance. Our results yield structural stability estimates for solutions of the moment-measure problem that are uniform over a natural class of convex functions, thereby addressing several questions that have been open in this direction.
A key novelty of our approach is that the Brascamp-Lieb stability bound is not only sharp in its stability exponent, but also uniform across a broad class of convex potentials. This uniformity is absent from previous results in the literature and, beyond its intrinsic mathematical interest, it is the mechanism that allows stability of the Brascamp-Lieb inequality to transfer to nonlinear variational problems such as the moment-measure problem. We moreover show that the $L^1-$nature of this stability estimate is sharp, in the sense that such a uniform estimate cannot hold in any $L^p$-metric, with $p>1$.