Sharp Quantitative Stability for the Pr\'ekopa-Leindler and Borell-Brascamp-Lieb Inequalities

Alessio Figalli; Marius Tiba; Peter van Hintum

REVIEW 3 major objections 5 minor 3 cited by

Sharp Quantitative Stability for the Pr\'ekopa-Leindler and Borell-Brascamp-Lieb Inequalities

T0 review · 3 major / 5 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read Near-equality in the Borell-Brascamp-Lieb inequality forces $f$ and $g$ to be $\sqrt{\delta}$-close, in $L^1$, to the same $p$-concave function.

desk verdict Claims to settle the sharp stability conjecture for PL and BBL, but the n-dimensional reduction has a gap for p near -1/n; the 1D and 2D parts are solid and worth reading. read the letter →

arxiv 2501.04656 v1 pith:UIE5ZVTT submitted 2025-01-08 math.FA math.APmath.COmath.MGmath.PR

Alessio Figalli , Peter van Hintum , Marius Tiba This is my paper · ORCID

classification math.FAmath.APmath.COmath.MGmath.PR MSC 52A4026B2539B62

keywords Borell-Brascamp-LiebinequalityPrékopa-LeindlerquantitativestabilityBrunn-Minkowskip-concavefunctionsL1approximationlevel-setmethodtransportmap

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 seeks to prove that the Borell-Brascamp-Lieb family of inequalities is sharply stable: when $f,g,h$ satisfy $h(\lambda x+(1-\lambda)y)\ge M_{\lambda,p}(f(x),g(y))$ with $\int f=\int g$ and $\int h=(1+\delta)\int f$, the functions $f$ and $g$ must be, up to translation, $O_{n,\lambda,p}(\sqrt{\delta})$ close in $L^1$ to a common $p$-concave function. This is the functional analogue of the sharp quantitative Brunn-Minkowski stability, and the $p=0$ case resolves the long-open sharp stability conjecture for the Prékopa-Leindler inequality, including log-concave functions in any dimension. The rate $\sqrt{\delta}$ is optimal, and the endpoint $p=-1/n$ is excluded because its equality class is genuinely richer. A reader should care because these inequalities encode how convexity and concavity behave under convolution-like operations, and the result says near-equality has a rigid, quantitatively controlled structure.

What carries the argument

The engine is a level-set and transport analysis. With $F_t=\{f>t\}$, $G_t=\{g>t\}$, $H_t=\{h>t\}$, a transport map $T$ defined by $dT/dt=|F_t|/|G_{T(t)}|$ connects corresponding levels, and Lemma 4.6 shows that outside a set of levels carrying $O(\delta)$ mass the map has derivative close to 1, the level sets are nearly convex, and $\lambda F_t+(1-\lambda)G_{T(t)}$ nearly coincides with $H_{M_{\lambda,p}(t,T(t))}$. This reduces the functional problem to a geometric one, where the sharp Brunn-Minkowski stability theorem is the black box that converts near-convexity into quantitative $L^1$ closeness. For Theorem 1.10 the machinery shifts to a variational maximizer $f'$ of the deficit and a 'shaving' argument: near every face of the $p$-concave hull, tilting the tangent $p$-plane and removing the cap must not pay off, and the stability of Brunn-Minkowski on the sets where equality holds forces the hull to be close.

What would settle it

Decisive test: check Theorem 1.1 itself. Construct equal-volume sets $A,B\subset\mathbb{R}^n$ with $|\lambda A+(1-\lambda)B|\le(1+\delta)|A|$ and compute the minimal $|K\setminus A|+|K\setminus B|$ over convex $K$ containing $A\cup B$; finding examples where this minimum is not $O(\sqrt{\delta})|A|$ would disprove the black box and with it the main theorem. A direct computation on the one-dimensional families in the paper also tests whether the $\sqrt{\delta}$ rate can be improved there.

Watch

Extended reading notes

Core claim

The central claim is Theorem 1.6: for every dimension $n$, every $\lambda\in(0,1/2]$, and every $p>-1/n$, if $f,g,h:\mathbb{R}^n\to\mathbb{R}_{\ge0}$ satisfy $\int f=\int g$, $h(\lambda x+(1-\lambda)y)\ge M_{\lambda,p}(f(x),g(y))$, and $\int h=(1+\delta)\int f$, then there exists a $p$-concave $\ell$ such that, up to translation, $\int(|f-\ell|+|g-\ell|)\,dx=O_{n,\lambda,p}(\sqrt{\delta})\int f\,dx$. The proof proceeds in two independent stages: Theorem 1.9 shows $f$ and $g$ are $\sqrt{\delta}$-close to each other, and Theorem 1.10 shows that when $f=g$, $f$ is linearly close in $\delta$ to its $p$-concave hull. The square-root rate is proved optimal, so the paper's goal is to show the whole Borell-Brascamp-Lieb range has exactly the stability that Brunn-Minkowski has.

Load-bearing premise

The proof rests on a black box: the sharp Brunn-Minkowski stability theorem the authors proved in two preprints. If that theorem contains an error, or if its hypotheses fail on the specific level sets constructed here, the whole bound collapses.

Editorial extensions

If this is right

For $p=0$, the theorem resolves the sharp Prékopa-Leindler stability conjecture, with a log-concave $\ell$ and optimal $\sqrt{\delta}$ rate in every dimension.
For every $p>-1/n$, near-equality in Borell-Brascamp-Lieb implies $L^1$ closeness of $f$ and $g$ to a common $p$-concave function at the same $\sqrt{\delta}$ rate as Brunn-Minkowski.
When $f=g$, the linear rate $\delta$ applies: the deficit $\int(M^*_{\lambda,p}(f,f)-f)\,dx$ controls $\int(\operatorname{cop}(f)-f)\,dx$ linearly.
The rate $\sqrt{\delta}$ is optimal, as shown by the one-dimensional families in the paper, and the endpoint $p=-1/n$ is necessarily excluded because its equality cases form a larger class.

Reading between the lines

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

Applied to other functional inequalities whose set-level analogue is Brunn-Minkowski-like, the same level-set transport decomposition might yield sharp stability without new geometric input.
The conjectured dependence $O_{n,p}(\sqrt{\delta/\lambda})$ on the interpolation parameter is stated as open in the paper; testing it would require refining the tube arguments, not changing the proof's architecture.
The linear self-stability of Theorem 1.10 suggests a general principle: for a single function, a small deficit in $M^*_{\lambda,p}(f,f)$ controls the distance to the $p$-concave hull, which could support sharp concentration estimates for $p$-concave measures.

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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

Colleague,

Read the Figalli–van Hintum–Tiba paper on sharp stability for Prékopa–Leindler and Borell–Brascamp–Lieb. The headline claim is exactly what it looks like: a unified proof of the √δ rate, resolving the conjecture for PL and extending to BBL. That matters if it's true, but I don't think the proof is complete as written.

The 1D and 2D cases look genuinely new and mostly solid. The 'shaving' variational argument in Section 9 is an interesting technique, and the paper is honest about the λ-dependence and the p→−1/n blow-up. It clearly supersedes the older non-sharp results.

The problem is in the n-dimensional reduction, §7.4. The reduction from R^n to R^2 produces functions f', g', h' satisfying a BBL-type inequality with parameter q = p/(1+np), and then invokes the 2D theorem, which requires q > −1/2. But q > −1/2 only when p > −1/(n+2). For p ∈ (−1/n, −1/(n+2)], q ≤ −1/2, so Theorem 6.1 doesn't apply. The overview in §3.3 explicitly promises q > −1/2 for every p > −1/n; that promise is false. This is not a minor technicality: it's an entire interval of p-values, including values like n=3, p=−0.3, where the claimed theorem should hold. As written, Theorem 1.6 is unproven for that range.

There's also the structural dependence on the authors' preprints [FvHT23, FvHT24] for sharp BM stability. That's fine if those preprints are correct, but the present paper's main theorem inherits any error there. The reader's report flagged this as the main weakness; I think the internal gap is the more serious issue.

Bottom line: the paper has substantial value in the 1D and 2D results and the new technique, but the main theorem as stated is not established. A serious referee should see it, but the authors need to either fix the reduction for p near −1/n or restrict the statement. I'd send it to review, with the expectation of major revision.

Referee Report

3 major / 5 minor

Summary. The paper claims a sharp quantitative stability theorem for the Borell–Brascamp–Lieb inequality in the full range p>-1/n (Theorem 1.6), with the Prékopa–Leindler case p=0 as a corollary. The proof is split into Theorem 1.9 (near-equality implies f and g are close in L1 up to translation, at order sqrt(delta)) and Theorem 1.10 (self-sup-convolution near-equality implies linear L1 closeness to a p-concave function). Theorem 1.9 is proved by a long chain of reductions: one-dimensional case, two-dimensional case, and then an n-dimensional slicing argument that reduces cones in R^n to two-dimensional problems. Theorems 1.10 is proved through a variational maximizer, p-face shaving, and reduction to a small-scale linear stability statement. The paper is structured carefully and is transparent about its limitations, including the excluded endpoint p=-1/n and the unoptimized dependence on lambda.

Significance. If correct, Theorem 1.6 is a definitive result: it gives the first sharp sqrt(delta) quantitative stability for Borell–Brascamp–Lieb in full generality and resolves the long-standing conjecture for Prékopa–Leindler. The paper correctly identifies the optimal exponent, excludes p=-1/n where the statement is false, and discusses the impossibility of requiring the approximating function to dominate f. The framework is coherent and builds on the authors' sharp Brunn–Minkowski stability, and the abstract and overview give a clear map of the many reductions. However, the n-dimensional reduction in Section 7.4 contains a concrete dimension mismatch that leaves a full interval of p-values unproved as written; the result is therefore currently incomplete, though the gap appears local and plausibly repairable. The manuscript is not machine-checked, and the proof is long enough that I could not independently verify every technical step.

major comments (3)

[Section 7.4, Lemma 7.16] The slices C_{z,w} in Definition 7.4 are (n-2)-dimensional, so the Brunn–Minkowski inequality applied to the inclusion C_{lambda(z,w)+(1-lambda)(z',w')} contains lambda C_{z,w}+(1-lambda) C_{z',w'} must use the exponent 1/(n-2), not 1/n. Lemma 7.16 instead invokes Lemma 7.17 with an inequality of the form b^{1/n} >= lambda a^{1/n} + (1-lambda) c^{1/n}, which is the wrong dimension. The correct parameter is q = p/(1+(n-2)p), not q = p/(1+np). With the written q, for every n >= 3 and every p in (-1/n, -1/(n+2)] the value q is <= -1/2 (for example n=3, p=-0.3 gives q=-3), so Theorem 6.1, which requires p > -1/2, cannot be applied. The overview in Section 3.3 promises a q > -1/2 for every p > -1/n; that promise is not delivered by the proof as written. This leaves the proof of Theorem 1.9, and hence of Theorem 1.6, incomplete for an entire interval of p-values.
[Section 7.4, Lemmas 7.16 and 7.17] Lemma 7.17 is stated only for p in (-1/(n+2),0), while Lemma 7.16 claims its conclusion for every p in (-1/n, infinity). Thus the proof of Lemma 7.16 does not cover p=0, which is exactly the Prékopa–Leindler case needed for Corollary 1.7, nor does it cover p>0. No separate argument is supplied for those cases. If the authors intend p=0 and p>0 to follow by limiting or monotonicity arguments, those arguments need to be written out; as it stands the reduction from Proposition 7.11 to Theorem 6.1 is not justified for a substantial part of the claimed parameter range. This issue is part of the same reduction as the previous comment and should be repaired together.
[Sections 4, 6, 7, and 9] The proof depends critically on Theorem 1.1 from the unpublished preprints [FvHT23, FvHT24], which are used as black boxes in Lemmas 4.6, 4.8, 6.6, 7.13 and in Section 9.3. If those preprints contain an error, or if their smallness hypotheses are not satisfied at any of the level sets constructed here, the present proof collapses. This is not an internal inconsistency, but it is a correctness risk. The authors should state precisely which version of Theorem 1.1 is being used and verify the hypotheses in each application, or update the references to the final published versions before publication.

minor comments (5)

[Section 5, Theorem 5.1] Theorem 5.1 is stated in R and the conclusion should be integral over R, but the displayed formula writes int_{R^2} |f-g| dx; this is a typo.
[Sections 1.3 and elsewhere] The name 'Borell-Brascamb-Lieb' appears in headings and in the abstract; it should be 'Borell-Brascamp-Lieb'.
[Section 7.4.1] In the proof after Lemma 7.16 there is a typo: 'there exist s v' should presumably be 'there exists v'.
[Section 7.4, Lemma 7.16] The first bullet point of Lemma 7.16 lacks an explicit quantifier; it should read 'for all x,y in C'' in the displayed inequality.
[Section 4, Lemma 4.9] The assumption involving eta co(F_{t0}) is stated with both o and v in eta co(F_{t0}); this should be clarified, in particular whether the condition is on v alone or on both points.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the functional stability results are derived from the independent set-level Brunn–Minkowski stability of the authors' previous work, not from the target statement.

full rationale

The central theorems 1.9 and 1.10 are proved by reducing to level-set and sup-convolution arguments that invoke Theorem 1.1, the sharp Brunn–Minkowski stability result from [FvHT23, FvHT24]. This is a self-citation and it is load-bearing, but it is not a circular reduction: Theorem 1.1 is a statement about sets, while Theorems 1.6, 1.9 and 1.10 are statements about functions. The paper never defines its target quantities in terms of Theorem 1.1, and no parameter is fitted to the claimed final estimate. The proof of Theorem 1.6 combines Theorem 1.9 (closeness of f and g) with Theorem 1.10 (linear closeness of f to a p-concave hull); the latter is established through an internally constructed variational maximizer in Section 9, not by re-using the theorem being proved. The one- and two-dimensional cases are proved essentially self-contained in Sections 5 and 6, and the n-dimensional case reduces to them. The flagged issue in §7.4, concerning the admissible range of q below −1/2 for p ∈ (−1/n, −1/(n+2)], is a correctness or rigor concern about the dimension reduction; it does not make the derivation equivalent to its inputs. The self-citation to [FvHT23, FvHT24] is independent support and therefore, under the given rules, does not raise the circularity score.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

The proof is analytic and does not introduce free parameters or invented entities. It rests on standard tools plus the authors' own sharp BM stability theorem, which is cited and used as a black box.

assumptions (7)

domain assumption Sharp quantitative stability of Brunn-Minkowski (Theorem 1.1 of [FvHT23,FvHT24])
Used as a black box in Lemmas 4.6, 4.8, 6.6, 7.13, and in Section 9.3 (via Theorem 1.1) to control level-set deviations.
standard math Brunn-Minkowski inequality itself
Used to relate |λF_t+(1−λ)G_t| to M_{λ,1/n}(|F_t|,|G_t|).
standard math Borell-Brascamp-Lieb inequality (Theorem 1.4)
Used when deriving lower bounds on ∫h and in fiberwise arguments.
standard math Knaster-Kuratowski-Mazurkiewicz lemma
Used in Lemma 4.4 to evenly partition functions among cones.
standard math John's ellipsoid theorem
Used in Lemma 7.14 to normalize cones.
standard math Zorn's lemma
Used in Proposition 9.10 to produce a minimal p-concave majorant.
domain assumption Theorem 7.9 from [FvHT23] on cone partitions
Used in Lemma 7.5 to construct fine cone decompositions.

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Cite this review

Pith. "Pith review of Sharp Quantitative Stability for the Pr\'ekopa-Leindler and Borell-Brascamp-Lieb Inequalities." pith.science (2026). https://pith.science/paper/UIE5ZVTT

@misc{pith2026250104656,
  author       = {Pith},
  title        = {Pith review of: Sharp Quantitative Stability for the Pr\'ekopa-Leindler and Borell-Brascamp-Lieb Inequalities},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/UIE5ZVTT}},
  note         = {Machine review of arXiv:2501.04656}
}

read the original abstract

The Borell-Brascamp-Lieb inequality is a classical extension of the Pr\'ekopa-Leindler inequality, which in turn is a functional counterpart of the Brunn-Minkowski inequality. The stability of these inequalities has received significant attention in recent years. Despite substantial progress in the geometric setting, a sharp quantitative stability result for the Pr\'ekopa-Leindler inequality has remained elusive, even in the special case of log-concave functions. In this work, we provide a unified and definitive stability framework for these foundational inequalities. By establishing the optimal quantitative stability for the Borell-Brascamp-Lieb inequality in full generality, we resolve the conjectured sharp stability for the Pr\'ekopa-Leindler inequality as a particular case. Our approach builds on the recent sharp stability results for the Brunn-Minkowski inequality obtained by the authors.

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Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.

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Stability of the pr \'e kopa--leindler inequality

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Stability of the Pr \'e kopa-Leindler inequality for log-concave functions

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Dorin Bucur and Ilaria Fragal \`a . Lower bounds for the Pr \'e kopa-Leindler deficit by some distances modulo translations. J. Convex Anal , 21(1):289--305, 2014

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A quantitative stability result for the Pr \'e kopa--Leindler inequality for arbitrary measurable functions

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Marco Barchiesi and Vesa Julin. Robustness of the G aussian concentration inequality and the B runn- M inkowski inequality. Calc. Var. Partial Differential Equations , 56, 05 2017

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Balogh and Alexandru Kristály

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On extensions of the Brunn-Minkowski and Prékopa-Leindler theorems, including inequalities for log concave functions, and with an application to the diffusion equation

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