Sharp Stability of Brenier Potentials and its Link to Poincaré Inequalities

The authors prove an inequality for uniformly log-concave measures that bounds the variance of a difference of convex functions and is stronger than the Poincaré inequality. The motivation is the stability of optimal transport: for a fixed uniformly log-concave source, it yields a sharp bound on how much the Brenier potential moves when the target measure is perturbed.

Applied to Gaussian measures, the inequality recovers the fact that the Bures-Wasserstein distance between two positive semidefinite matrices \(A\) and \(B\) is equivalent to the Frobenius distance \(\|A^{1/2} - B^{1/2}\|_F\). The proofs linearise the Monge-Ampère equation that governs quadratic optimal transport. Stability bounds of this kind are the basic input for statistical rates when transport maps are estimated from samples.