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Math-2837 · mathematics · Pde

The optimal one-third stability exponent for compact-target Brenier maps

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
Fix $d\geq 2$, a convex body $K\subset R^d$ with nonempty interior, its uniform probability rho, and a compact target set Y. Is there C(K,Y) such that $||T_\mu -T_\nu ||_(L2(\rho ))\leq C$ $W_2(\mu ,\nu )^{1/3}$ for every pair of target probability measures on Y, including atomic and singular measures? Is no larger universal exponent possible even for cube source and target and exactly three target atoms?

Short English statement

Fix $d\geq 2$, a convex body $K\subset R^d$ with nonempty interior, its uniform probability rho, and a compact target set Y. Is there C(K,Y) such that $||T_\mu -T_\nu ||_(L2(\rho ))\leq C$ $W_2(\mu ,\nu )^{1/3}$ for every pair of target probability measures on Y, including atomic and singular measures? Is no larger universal exponent possible even for cube source and target and exactly three target atoms?

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2837-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2837

Sharp one-third Brenier stability: The source is the uniform measure of one fixed convex body, and all targets share one compact support bound. Atomic targets are allowed; the three-atom construction rules out every exponent above $1/3$ in this uniform setting.

With the hypotheses and conventions above, the claimed result addresses the whole mathematical assertion on this page.

The optimal one-third stability exponent for compact-target Brenier maps

OpenAI's claim

Sharp one-third Brenier stability: The paper claims a uniform $L^2$ stability estimate with exponent $1/3$ for quadratic optimal maps from a uniform convex-body source to targets in a fixed compact set. A three-atom family shows that larger exponents, including the proposed square-root estimate, fail.

Mathematical statement of the claimed result

Mathematical claim 1.
Fix $d\ge2$, a compact convex $K\subset\mathbb R^d$ with nonempty interior, uniform source probability $\rho=\mathbf1_Kdx/|K|$, and nonempty compact target set $Y$. For the quadratic-cost Brenier maps $T_\mu$ to arbitrary probability measures $\mu$ on $Y$, the claimed uniform stability is \[\|T_\mu-T_\nu\|_{L^2(\rho)}\le C(K,Y)W_2(\mu,\nu)^{1/3}.\] Here $W_2$ minimizes the root-mean-square distance over couplings. Atomic, singular and absolutely continuous targets are all permitted. The exponent is asserted optimal: no larger exponent works uniformly, even with $K=Y=[-1,1]^d$ and three-atom targets.

Outline of the argument

Sharp one-third Brenier stability: In the obstruction, tilting one plane in a three-plane maximum switches a thin source cell between separated target slopes while preserving its mass. For the upper estimate, semi-discrete cell calculus tracks cell masses and first moments as slopes and intercepts vary. The central moment comparison controls the map difference, and approximation extends the estimate to arbitrary compactly supported targets.

Formal verification

Family 374 describes Lean coverage for uniform one-third map stability and its optimality through three-atom targets.

Sharp one-third Brenier stability: The manuscript links Brenier.lean.

The documented selection includes this manuscript’s main assertion; that assertion retains the restrictions stated above.

This site has not run the Lean development. These links describe OpenAI’s selected formal statements and should be read with their stated scope.

Status caveat

OpenAI reports that an internal model produced the manuscripts. Their arguments have not been independently verified as proofs by this site. Any portion outside the documented formal scope may contain mistakes, and the catalogue label reports the announced mathematical claim.

Sources

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