OpenAI research collection · 1 manuscript · Source checked 2026-10-19
OpenAI's result summary
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)}≤ 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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MANUSCRIPT 01
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…
Full source abstract
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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New mathematical reasoning could help AI tackle harder scientific and computational problems. How reliably these skills transfer beyond mathematics will determine how much they contribute to general intelligence.