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Sharp one-third stability of Brenier maps

For uniform source measure ρ on a compact convex body with interior in dimension at least two, quadratic optimal transport maps satisfy

\|T_\mu-T_\nu\|_{L^2(\rho)}\le C W_2(\mu,\nu)^{1/3}

uniformly over targets in a fixed compact set. The exponent is sharp, even for three-atom targets, disproving Letrouit's conjectured square-root bound.

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Sharp One-Third Stability of Brenier Maps

Sep 25, 2026·article.pdf

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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