Sharp One-Third Stability of Brenier Maps

For the uniform probability measure ρ on a compact convex body in ℝ${}^d$, d ≥ 2, quadratic optimal transport maps are one-third Hölder continuous in $L^2(\rho )$ with respect to the target's 2-Wasserstein distance. The constant is uniform over all targets supported in a fixed compact set, and the exponent is optimal. A three-atom family on a fixed cube disproves Letrouit's conjectured uniform square-root estimate.