Quantitative stability for the Brascamp-Lieb inequality and moment measures

Jo\~ao Miguel Machado; Jo\~ao P.G. Ramos

REVIEW 4 major objections 4 minor 13 references

Quantitative stability for the Brascamp-Lieb inequality and moment measures

T0 review · 4 major / 4 minor · reviewed 2026-08-03 · deepseek-v4-flash

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Pith's one-line read This paper proves a quantitative stability theorem for the Brascamp-Lieb inequality: any function with small deficit is, in L1, within a dimension-only constant times the square root of the deficit of the manifold of affine optimizers.

desk verdict A real advance on Brascamp-Lieb stability and moment measures, but the proof leans on an unverified PL black box and has a couple of fixable slips; worth refereeing, not desk-rejecting. read the letter →

arxiv 2511.22636 v3 pith:BFDNO6ZI submitted 2025-11-27 math.FA

João Miguel Machado , João P.G. Ramos This is my paper · ORCID

classification math.FA MSC 39B8226D1049Q22

keywords Brascamp-LiebinequalityquantitativestabilitymomentmeasuresPrékopa-LeindlerL1distancelog-concaveoptimaltransportWasserstein

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper establishes the first quantitative stability theorem for the Brascamp-Lieb variance inequality whose constant depends only on the ambient dimension, not on the underlying convex potential. It shows that if a function almost attains the inequality under a Gibbs measure proportional to e^{-φ}, then its L1 distance to the finite-dimensional manifold of optimal functions {a·∇φ+b} is bounded by a universal dimension constant times the square root of the deficit. The square-root rate is shown to be sharp, and a uniform estimate of this kind in any Lp metric with p>1 is impossible. Because the constant is uniform over all essentially continuous, normalizable convex φ, the same mechanism transfers to moment measures: small gaps in the variational functional J_μ imply L1 closeness of Gibbs measures and potentials to the optimal sets, giving stability for moment-measure representations and explicit α^{1/2} convergence of regularized moment measures.

What carries the argument

The load-bearing object is the Brascamp-Lieb deficit δ_BL(f), paired with the finite-dimensional manifold of optimizers O_BL={a·∇φ+b}. The proof runs a linearization of the deficit through the Prékopa-Leindler inequality with parameter 1/2, and the sharp input is a stability estimate for that inequality that converts an ε-deficit into an L1 error of order ε^{1/2} with a dimension-only constant. For moment measures, the same logic runs through the concave functional J_μ(φ)=log∫e^{-φ*} - ∫φ dμ, whose second variation is minus the Brascamp-Lieb deficit and whose optimizers are exactly the moment-measure potentials.

What would settle it

Find an essentially continuous convex φ and a sequence f_n with δ_BL(f_n)→0 such that dist_{L1(ϱ_φ)}(f_n,O_BL) grows faster than C_d δ_BL(f_n)^{1/2} for any dimension-only C_d; or construct a counterexample to the imported Prékopa-Leindler stability estimate at s=1/2, i.e., functions f,g,h satisfying the Prékopa condition with relative mass gap ε→0 whose minimal L1 distance to a common log-concave profile tends to zero more slowly than ε^{1/2} with a dimension-independent constant.

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Extended reading notes

Core claim

On its own terms, the paper's central claim is Theorem 1.1: for any essentially continuous convex φ:R^d→R∪{+∞} with ∫e^{-φ}<∞, there is a constant C_d depending only on d such that dist_{L1(ϱ_φ)}(f, {a·∇φ+b}) ≤ C_d δ_BL(f)^{1/2} for all locally Lipschitz f∈L²(ϱ_φ). Here δ_BL(f) is the Brascamp-Lieb deficit, the nonnegative gap between the curvature-weighted energy ∫⟨(D²φ)^{-1}∇f,∇f⟩ dϱ_φ and the variance Var_{ϱ_φ}(f). The proof obtains this by linearizing through the Prékopa-Leindler inequality and then invoking a sharp quantitative Prékopa-Leindler stability estimate; the uniformity is what lets the estimate transfer to nonlinear variational problems. The companion moment-measure results sh

Load-bearing premise

The proof relies on a sharp quantitative Prékopa-Leindler stability estimate, imported from another work, whose constant is asserted to depend only on the dimension at the interpolation parameter 1/2; if that constant actually depends on the functions' masses or higher moments, the uniformity of the main theorem collapses.

Editorial extensions

If this is right

Near-equality in the Brascamp-Lieb inequality is equivalent, up to a square root, to L1 closeness to affine-in-gradient functions, with the same constant for every essentially continuous convex potential; this makes the estimate usable when the potential is unknown or varying.
Stability transfers to the moment-measure problem: a small gap in J_μ forces the corresponding Gibbs measure to be close in L1 to the optimal moment-measure manifold, yielding a Polyak-Lojasiewicz-type inequality.
Regularized moment measures converge to their classical counterparts at rate α^{1/2} in L1, with constants depending only on the dimension and on a moment of the target.
Over R^d, in classes with either a uniform directional spread or a uniform Hessian bound, distances between moment-measure optimal sets are controlled by a fractional power of W_2(μ,ν), with exponent (p-2)/(6p-4).
No uniform estimate of this form can hold in Lp for p>1, so L1 is the natural metric for this stability theory.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The dimension-only uniformity suggests the Brascamp-Lieb stability estimate can be used as a black box in numerical schemes for moment measures, giving convergence certificates that do not require knowing the target's convex potential.
The sharp 1/2 exponent in L1 matches the Brunn-Minkowski-type stability pattern; one might expect analogous uniform stability for other functional inequalities proved through Prékopa-Leindler, with different interpolation parameters.
If the imported Prékopa-Leindler stability estimate were replaced by a fully self-contained proof, the method would be unconditional; that replacement is a testable boundary of the paper's contribution.
The impossibility of uniform Lp stability for p>1 implies that any quantitative stability in stronger metrics must necessarily depend on the potential, so the L1 framework likely sets the right scale for applications.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The short version: the paper delivers the first φ-uniform L¹ stability estimate for the Brascamp-Lieb inequality, and the applications to moment measures are natural and mostly solid. But the main theorem is only as secure as the Prékopa-Leindler black box it imports, and there are a couple of spots in the writing that a referee should force the authors to clean up before I'd take the details on faith.

What's actually new: Theorem 1.1 is a genuine advance. The Bobkov-Ledoux linearization is a known route, but combining it with the sharp PL stability to get a dimension-only constant independent of φ is new, and the transfer to moment measures (Theorems 1.2–1.4) looks like the right way to get quantitative stability there. I believe the core argument is basically sound.

The soft spots, in order of importance. First, Lemma 2.7 is imported from [FvHT25] and the proof of the lemma has an algebraic slip: dividing (2.20) by the masses gives different prefactors for f and g. The fix is to apply the black box to the normalized triple f/∫f, g/∫g, h/√(∫f∫g), but as written the step doesn't work. More importantly, the whole theorem is conditional on a result in a preprint you don't prove. That's not disqualifying, but it makes the uniformity claim a little fragile. Second, the bootstrap in Theorem 4.11 has a real arithmetic inconsistency: the alternating S_k doesn't match the recurrence from Caffarelli. Iterating correctly gives Lip(∇φ_α) → Λ, not Λ^{1/3}. The theorem may still be true, but the proof as written is wrong. Third, the abstract claims no Lp estimate can hold for p>1, but the text never states or proves that. That's an unfulfilled promise.

The reader's take is about right—conditional, with the main concern being the black box. I'd add the 4.11 arithmetic issue as a concrete thing to fix. The paper is for specialists in optimal transport and functional inequalities. It deserves a serious referee, but only if the referees require the authors to fix the normalization in Lemma 2.7, correct the bootstrap, and either prove the Lp sharpness or cut it from the abstract.

My recommendation: send it to peer review, but ask for a careful revision.

Referee Report

4 major / 4 minor

Summary. The paper develops a quantitative stability theory for the Brascamp–Lieb variance inequality and for moment measures. The main result, Theorem 1.1 (restated as Theorem 3.2), asserts that for every essentially continuous convex potential φ with ∫e^{-φ}<∞ there is a dimension-only constant C_d such that the L^1(ϱ_φ)-distance from any locally Lipschitz f∈L^2(ϱ_φ) to the manifold of optimizers O_BL={a·∇φ+b} is bounded by C_d δ_BL(f)^{1/2}. The proof follows Bobkov–Ledoux's linearization strategy, with the key quantitative input being a sharp stability version of the Prékopa–Leindler inequality imported from the recent preprint [FvHT25] (Lemma 2.7). The authors then apply Theorem 1.1 to obtain stability estimates for moment measures: Theorem 1.2 gives a Polyak–Łojasiewicz-type estimate for the functional J_μ; Theorem 4.2 gives stability on compact domains; Theorem 4.4 gives convergence rates for quadratically regularized moment measures; and Theorems 4.7 and 4.13 give W_2-stability over two classes of measures, K_ϑ and K_Λ. The paper also claims that the L^1 nature of the BL stability is sharp, in that no uniform L^p estimate for p>1 can hold.

Significance. If the main results are correct, this is a substantial contribution. A quantitative Brascamp–Lieb stability estimate with a dimension-only constant uniform over all essentially continuous convex potentials is new, as is the transfer of such uniformity to stability of moment measures. The explicit convergence rates for regularized moment measures and the W_2-stability bounds for the classes K_ϑ and K_Λ are concrete and potentially useful in numerical applications. The proof strategy is natural and the manuscript is generally clearly written. The paper also correctly identifies that the load-bearing input is the sharp PL-stability lemma from [FvHT25]; if that external result is valid with the stated constants, Theorem 1.1 follows. However, as detailed below, the proof contains several gaps that need to be addressed before the claims can be accepted.

major comments (4)

[§3, proof of Theorem 3.2 (cluster point of x_δ/2δ)] The argument that x_δ/2δ has a cluster point as δ→0+ is not fully justified. The proof uses the sets Ω_n={D^2φ≤n id} and derives uniform bounds on ∫_{Ω_n}|(x_δ/2δ)·∇φ|dϱ_φ. It then claims, via dominated convergence and convergence of 1_{Ω_n} to 1 in L^1(ϱ_φ), that limsup over n of the integrals equals ∫|θ̄·y|dμ_φ(y) times limsup ||x_δ||/(2δ). This step is not valid as stated: the integrand |(x_δ/||x_δ||)·∇φ| is not dominated by an L^1 function, the direction (x_δ/||x_δ||) varies with δ, and monotone convergence applies to fixed nonnegative functions, not to a sequence with varying directions. Lemma 2.3 gives inf_θ∫|θ·y|dμ_φ(y)>0 for the full integral, but the proof does not control the contribution from the complement of Ω_n or the variation of the directions. This gap is load-bearing because the boundedness of x_δ/δ is essential for extracting a cluster point and hence for the final L^1
[§2.3, Lemma 2.7 and Theorem 3.2 (external black box)] The uniformity claim of Theorem 1.1 rests entirely on Lemma 2.7, which is imported verbatim from the very recent preprint [FvHT25]. The proof of Lemma 2.7 is only a sketch: it states that estimates (2.20) hold with α=1/2 and a dimension-only constant, but it does not verify that the constant in [FvHT25] is independent of the masses or higher moments of f,g after the normalization to probability densities. The displayed formulas in Lemma 2.7 do use a common prefactor 1/(∫f∫g)^{1/2}, so the particular prefactor concern raised by the stress-test does not appear in the current text; however, the underlying issue remains. If the external theorem's constant secretly depends on ∫f, ∫g, or on higher moments, the normalization step would introduce extra factors and the dimension-only constant in Theorem 1.1 would fail. The manuscript should either give a self-contained proof of Lemma 2.7 or state
[§4.3, proof of Theorem 4.11 (arithmetic inconsistency in the bootstrap)] The bootstrap argument for Λ^{-1/3}-strong convexity of the moment measure potential is arithmetically inconsistent. Starting from Lip(∇φ_α) ≤ sqrt(Λ/α), the next iteration using Caffarelli's contraction theorem should yield an exponent S_2=3/4 for Λ (i.e., Lip(∇φ_α) ≤ Λ^{3/4}/α^{1/4}), not S_2=1/4 as given by the displayed recurrence S_k = Σ_{i=1}^k (-1)^{i+1}/2^i. More fundamentally, if the induction follows the contraction theorem, the recurrence for the exponent of Λ is S_{k+1}=(1+S_k)/2 with S_1=1/2, whose limit is 1, not 1/3. To obtain a limit of 1/3 one would need S_{k+1}=(1-S_k)/2, which is not what the contraction theorem gives. Thus the proof as written does not establish the claimed Λ^{-1/3}-strong convexity. Since Theorem 4.13 (and hence Theorem 1.4 for the class K_Λ) depends on this regularity result, the stability claim for K_Λ is not proven.
[§3, approximation step (3.5)–(3.6)] The reduction from α-strongly convex potentials to general essentially continuous convex φ is sketched too tersely. For (3.5), the passage to the limit inside ∫⟨(D^2φ)^{-1}∇f,∇f⟩dϱ_{φ_α} requires more than pointwise convergence of the measures; one needs an argument controlling the possible concentration of the integrand as α→0. For (3.6), the claim that the optimal a_α lie in a fixed compact ball relies on equibounded second moments of ϱ_{φ_α}, which is plausible but not explicitly shown. These steps are probably repairable, but they are load-bearing for the full generality of Theorem 1.1 and should be written out carefully.

minor comments (4)

[§4, proof of Theorem 4.1 (second estimate)] In the line after the Taylor expansion, the inequality 'inf_{a,b} ||(φ-φ̄)-(a·x+b)||^2_{L1(μ_{φ*_λ})} ≤ 2C_d (Jμ(φ̄)-Jμ(φ))^{1/2}' is dimensionally inconsistent: the left-hand side is a squared L1 distance, so the right-hand side should be a multiple of the energy difference, not its square root. The mean-value argument gives 2C_d^2 (Jμ(φ̄)-Jμ(φ)). The statement of Theorem 1.2 remains plausible up to constants, but the proof as written needs correction.
[§2, Theorem 2.4 and elsewhere] There are numerous typographical errors and small inconsistencies: 'mesuare' for 'measure', 'Stabiliy' in the header, 'interwined' for 'intertwined', 'the deficit if upper semi-continuous' should be 'the deficit is upper semi-continuous', and the missing factor 1/2 in the expression for E_{μ,α}(ϱ)-E_μ(ϱ) in the proof of Theorem 4.4. These do not affect the mathematics but should be fixed.
[§4.1, Remark 4.3] The remark states that for α>0 the argument via quantitative stability of Brascamp-Lieb does not apply, but the discussion is brief. It would help to clarify whether the issue is the lack of the appropriate stability estimate for the regularized deficit, or a separate obstruction.
[§1, Theorem 1.4] The theorem states 'for each p>2 there exists a constant C such that ... C grows linearly in p.' The dependence on p is discussed later, but the statement would benefit from making explicit that C also depends on Ω, ϑ or Λ and d, as done in Theorems 4.7 and 4.13.

Circularity Check

0 steps flagged · score 1.0 of 10

No circularity: the main Brascamp-Lieb stability bound is reduced to an external sharp Prékopa-Leindler stability result, not to fitted data or load-bearing self-citations.

full rationale

The central result, Theorem 1.1 / Theorem 3.2, follows the Bobkov-Ledoux linearization: setting u_delta = exp(2 delta f - phi), v_delta = exp(-phi), w_delta = exp(f_delta - phi), the paper expands the Prékopa-Leindler deficit as epsilon_delta = (1+O(delta)) (delta^2/2)(delta_BL(f)+o(delta^2)/delta^2) (Section 3, Eqs. (3.2)-(3.3)). The only quantitative input is Lemma 2.7, imported from [FvHT25], whose authors are not the present authors. Applying that external sharp PL-stability estimate gives closeness of u_delta to e^{-phi(·+x_delta)} in L1, and the subsequent cluster-point argument converts this into dist_{L1(rho_phi)}(f,O_BL) ≤ C_d delta_BL(f)^{1/2}. Nothing is fitted to f or phi, and the target distance is not used to define the deficit. The moment-measure results in Section 4 are obtained by applying this BL stability to the second variation of J_mu (Lemma 2.5 and Theorem 4.1) and by combining it with external results from [CEK15], [San16], [DF25], [Caf92], and Klartag's Lemma 4.6; they do not feed back into the proof of Theorem 3.2. Self-citations [BFR23], [FR24], [BFR25] appear only in the literature survey and are not load-bearing; in particular [FR24] is cited for context but the proof uses [FvHT25] instead. I flag that Lemma 2.7 is an imported black box and its normalization step is terse, but this is an external-verification or robustness concern, not circularity: if the imported lemma failed, the theorem would be unsupported, but the derivation would not be identifying its conclusion with its hypotheses. There is no circular step to report.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

No fitted parameters or invented entities. The results rest on a bundle of external theorems: sharp Prékopa-Leindler stability (FvHT25), Caffarelli's contraction theorem, the Cordero-Erausquin-Klartag existence theory for moment measures, and Klartag's growth estimate. These are standard within the field but unproved in this paper, and the Prékopa-Leindler stability is very recent.

assumptions (5)

domain assumption Sharp Prékopa-Leindler stability from [FvHT25] (Lemma 2.7): if ∫h ≤ (1+ε)(∫f∫g)^{1/2}, then ∫ |f/∫f - g(·+x0)/∫g| dx ≤ C_d ε^{1/2}.
External unproved result used as a black box; load-bearing for Theorem 1.1.
domain assumption Caffarelli's contraction theorem [Caf92,Caf00]: if D²V≤Λ id and D²W≥λ id, the optimal transport map has Lipschitz constant sqrt(Λ/λ).
Used in the bootstrap proof of Theorem 4.11.
domain assumption Existence, uniqueness, and essential-continuity theory for moment measures [CEK15, San16].
Used throughout Section 4 to define moment measures and their variational characterizations.
domain assumption Klartag's growth estimate [Kla14, Lemma 2] for Lipschitz convex functions with a lower spread bound.
Used in Theorem 4.7 to uniformly bound moments of Gibbs measures in the class K_ϑ.
standard math Standard optimal transport duality, geodesic convexity of entropy and maximal correlation, and the interpolation inequality Lemma 2.1.
Background tools used throughout the paper.

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Pith. "Pith review of Quantitative stability for the Brascamp-Lieb inequality and moment measures." pith.science (2026). https://pith.science/paper/BFDNO6ZI

@misc{pith2026251122636,
  author       = {Pith},
  title        = {Pith review of: Quantitative stability for the Brascamp-Lieb inequality and moment measures},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/BFDNO6ZI}},
  note         = {Machine review of arXiv:2511.22636}
}

abstract

We develop a quantitative stability theory for moment measures based on a new sharp uniform stability principle for the Brascamp-Lieb variance inequality in terms of the $L^1$-distance. Our results yield structural stability estimates for solutions of the moment-measure problem that are uniform over a natural class of convex functions, thereby addressing several questions that have been open in this direction. A key novelty of our approach is that the Brascamp-Lieb stability bound is not only sharp in its stability exponent, but also uniform across a broad class of convex potentials. This uniformity is absent from previous results in the literature and, beyond its intrinsic mathematical interest, it is the mechanism that allows stability of the Brascamp-Lieb inequality to transfer to nonlinear variational problems such as the moment-measure problem. We moreover show that the $L^1-$nature of this stability estimate is sharp, in the sense that such a uniform estimate cannot hold in any $L^p$-metric, with $p>1$.

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Reference graph

Works this paper leans on

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Pith tools

Reviewed August 3, 2026 · model on record in the stance chip above.

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Sharp stability of the brunn-minkowski inequality via optimal mass transportation.arXiv preprint arXiv:2407.10932,

[FvHT24] Alessio Figalli, Peter van Hintum, and Marius Tiba. Sharp stability of the brunn-minkowski inequality via optimal mass transportation.arXiv preprint arXiv:2407.10932,

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[Kla14] Bo’az Klartag. Logarithmically-concave moment measures i. InGeometric Aspects of Functional Anal- ysis: Israel Seminar (GAFA) 2011-2013, pages 231–260. Springer,

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Gluing methods for quantitative stability of optimal transport maps.arXiv preprint arXiv:2411.04908,

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arXiv

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arXiv

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1983