Sharp one-third stability of Brenier maps

What OpenAI claims

For uniform source measure ρ on a compact convex body with interior in dimension at least two, quadratic optimal transport maps satisfy ‖T_μ-T_ν‖_L²(ρ) ≤ C W₂(μ,ν)^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)

Paper

1. Sharp One-Third Stability of Brenier Maps

   25 September 2026 · Preprint, not peer reviewed · PDF

   OpenAI's abstract For the uniform probability measure ρ on a compact convex body in ℝd, d ≥ 2, quadratic optimal transport maps are one-third Hölder continuous in L²(ρ) 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.

Status

Preprint. The paper has not been independently reviewed. Treat the result as OpenAI's claim until other mathematicians have checked it. If a video exists, it illustrates the statement; it is not the proof.