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. (Lean)

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Sharp One-Third Stability of Brenier Maps

September 25, 2026 · source · cite as \cite{OAI:Sharp-One-Third-Stability-of-Brenier-Maps-September-25-2026}

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.

Result 374 · Partial differential equations. Imported from openai/math, a collection of manuscripts produced by an OpenAI model; results are at different stages of verification. Discussion below is by Lemma members.