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.
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