Result 374, Partial differential equations

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 formalization New or sharp bound

The bigger picture

Why it matters

Optimal transport finds a map that moves one probability distribution to another while minimizing average squared travel distance. This result identifies exactly how sensitive that map can be to changes in the destination distribution.

What changes?

The manuscript reports this bound for a uniform source on any compact convex body with nonempty interior in dimension at least two. For all target distributions supported in a fixed compact set, the root-mean-square difference between their optimal maps is at most a constant times the cube root of their 2-Wasserstein distance. That distance is the least root-mean-square transport cost between targets. The constant is uniform over those targets, but may depend on the fixed source and support set.

What does that help mathematicians do?

The claimed sharpness rules out any larger exponent in a uniform estimate under these assumptions, including Letrouit's conjectured square-root bound. The manuscript reports that targets concentrating mass at just three points already exhibit the obstruction on a fixed cube. Thus, limiting targets to very few points does not remove the worst-case sensitivity. Researchers seeking stronger bounds would need additional restrictions that exclude this example.

Are there practical applications?

Its immediate value is foundational: it quantifies how accurately transport maps follow approximations of a target distribution. If approximating targets stay in the same fixed compact set and converge in 2-Wasserstein distance, their maps converge in root-mean-square error at the stated rate. This provides an error guarantee for such approximations, not a numerical algorithm or a demonstrated computational speedup.

This section was generated by GPT-6 Astra Medium. This explanation is based on the result summary and manuscript abstracts below. This context is separate from OpenAI's source text.

Manuscript

Sharp One-Third Stability of Brenier Maps

September 25, 2026 18 pages

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.

PDF LaTeX source

Cite (BibTeX)

@misc{OAI:Sharp-One-Third-Stability-of-Brenier-Maps-September-25-2026,
  author = {{OpenAI}},
  title = {{Sharp One-Third Stability of Brenier Maps}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Sharp-One-Third-Stability-of-Brenier-Maps-September-25-2026/article.pdf}{OAI:Sharp-One-Third-Stability-of-Brenier-Maps-September-25-2026}},
  year = {2026}
}

Lean formalization

OpenAI's note on what the formalization covers, from lean/docs/374.md.

Sharp one-third stability of Brenier maps

The following describes the scope of the Lean formalization related to the following accompanying paper(s):

Sharp One-Third Stability of Brenier Maps

Scope

The formalization proves one-third Hölder stability of Brenier maps from the uniform measure on a compact convex body with nonempty interior in $\mathbb R^d$ , for every $d\ge2$ . For targets supported in one fixed nonempty compact set, the $L^2$ distance between the unique quadratic optimal maps is at most a constant times the one-third power of the targets' $2$ -Wasserstein distance. The constant is uniform over those targets.

It also proves optimality of the exponent: on a fixed cube, pairs of three-atom target measures violate every analogous bound with exponent greater than $1/3$ . In particular, the conjectured uniform square-root estimate fails.

Comparator links

Result	Comparator statement
One-third stability of Brenier maps and sharpness	Brenier.lean

At a glance

Subject: Partial differential equations
Manuscripts: 1
Pages: 18
Dated: Sep 25, 2026
Lean formalization: Yes
Reasoning summary: No

Named problems addressed

Letrouit's conjectured uniform square-root estimateDisproved

Indexed by Emergent Mind with an LLM from the text above; check the manuscripts for the precise statements.

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On the map

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