Sharp one-third stability of Brenier maps

1 · The essentials

Imagine spreading material evenly through a solid shape, then moving it to a chosen destination pattern as cheaply as possible, with longer moves charged by the square of their distance. This work asks how much the resulting movement plan can change when the destination pattern changes. The starting shape must be compact and convex, have an interior, and lie in dimension two or higher. All destination patterns must stay inside one fixed compact set. Under these conditions, the change in the movement plan is bounded by a constant times the cube root of the change in the destination pattern, using the measurements specified below. The manuscript also shows that no larger power can work uniformly: even destination patterns concentrated at just three points rule out the previously conjectured square-root bound. The main result has a Lean formalization, although the supplied source does not specify its coverage of the individual claims.

A picture to help (an imperfect image, not to be taken literally)

As an imperfect image, imagine assigning material from an evenly filled tray to collection bins while minimizing the total squared travel distance. Moving the bins or changing how much each must receive can alter which parts of the tray supply each bin; the result limits how much the whole assignment changes, rather than tracking any single piece.

Why it matters

A stability estimate connects changes in the desired destination to changes in the optimal transport map. Here the manuscript identifies the best possible power for a bound uniform over the stated target class. Its three-point example also rules out Letrouit's conjectured uniform square-root estimate under these conditions.

2 · Going further

Let $ρ$ be the uniform probability measure on a compact convex body with nonempty interior in $R^{d}$ , where $d \geq 2$ . For target probability measures $μ$ and $ν$ supported in a fixed compact set, let $T_{μ}$ and $T_{ν}$ be their quadratic optimal transport maps from $ρ$ , called Brenier maps. The stability problem is to control the distance between these maps by the distance between their targets.

The main estimate is $∥ T_{μ} - T_{ν} ∥_{L^{2} (ρ)} \leq C W_{2} (μ, ν)^{1/3}$ . Here the left side measures the root-mean-square difference of the maps over the source, and $W_{2}$ is the targets' 2-Wasserstein distance. The constant is uniform over targets supported in the fixed compact set. Thus the estimate is one-third Hölder continuity of the transport map with respect to its target.

The manuscript states that the exponent is optimal. It supplies a three-atom family on a fixed cube that rules out Letrouit's conjectured uniform square-root estimate. The supplied abstract presents the upper bound and this sharpness example, but does not describe the argument establishing the upper bound or give the construction of the example. Only one manuscript is supplied, so no relationship with companion manuscripts can be described.

Useful notions

Probability measures and their support
Compact convex bodies
Quadratic optimal transport
Root-mean-square and L² distances
Hölder continuity

What was known before / what is new

Before

The source identifies a conjecture of Letrouit proposing a uniform square-root stability estimate. It supplies no earlier stability bounds or further historical context.

New

The manuscript establishes a uniform one-third stability estimate for the stated source and target classes. It also shows that the exponent is optimal, with a three-atom family on a fixed cube ruling out the conjectured square-root estimate. The main result has a Lean formalization.

Limits and assumptions

The estimate assumes a uniform source on a compact convex body with nonempty interior in dimension at least two, quadratic transport cost, and targets supported in a fixed compact set. Sharpness concerns the uniform bound over this class; the supplied source does not address improved estimates for more restricted targets or other source measures. The main result has a Lean formalization, but the supplied information does not specify whether that formalization includes the sharpness example or every manuscript claim. No remaining open problems are identified in the supplied text.

Mini glossary

Brenier map: In this setting, a map that transports the source probability measure to a target while minimizing the average squared travel distance.
2-Wasserstein distance: A distance between probability measures obtained from the minimum average squared transport cost by taking its square root.
L² distance: Here, the square root of the average squared difference between two transport maps, measured over the source.
One-third Hölder continuity: A bound in which the change in the output is at most a constant times the cube root of the change in the input.
Three-atom target: A target probability measure whose mass is concentrated at three points.
Sharp exponent: An exponent that cannot be replaced by a larger one while retaining the stated uniform estimate.

Keywords : Brenier maps · Optimal transport stability · Sharp Hölder exponent

AI-generated explanation based on the manuscript. Unofficial. Refer to the manuscript for the authoritative version.

3 · Technical details and sources

Claim made in the manuscript (source text)

For uniform source measure ρ on a compact convex body with interior in dimension at least two, quadratic optimal transport maps satisfy $∥ T_{μ} - T_{ν} ∥_{L^{2} (ρ)} \leq C W_{2} (μ, ν)^{1/3}$ uniformly over targets in a fixed compact set. The exponent is sharp, even for three-atom targets, disproving Letrouit's conjectured square-root bound.

Verified by Lean: the Lean software checked the part of the reasoning described below, sometimes only part of the manuscript. This does not mean independent mathematicians have validated the result.

Scope of the Lean formalization (source text)

…

lean/docs/374.md

Manuscripts in this family

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Sharp One-Third Stability of Brenier Maps

25/09/2026

To be verified

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 $L^{2} (ρ)$ 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.

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