{"id":"03a7ba0f-531a-4f4b-9b15-0165aac4f284","arxiv_id":"2501.04656","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Sharp quantitative stability for the Borell-Brascamp-Lieb inequality (and hence Prékopa-Leindler) is proven: near-equality of the integral implies an O(√δ) L1-distance to a p-concave function.","lead":"This paper proves that if two functions are nearly optimal for the Prékopa-Leindler or Borell-Brascamp-Lieb inequalities, they must be close to a single p-concave function, with the optimal square-root error. It resolves the sharp stability conjecture for these foundational inequalities, building on the authors' prior sharp stability result for Brunn-Minkowski.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Dimension mismatch in §7.4: the reduction from R^n to R^2 uses BM exponent 1/n on (n−2)-dimensional slices and q=p/(1+np), which falls below −1/2 for p∈(−1/n,−1/(n+2)], so Theorem 6.1 cannot be applied.","rationale":"The reader's weakest assumption is the dependence on the unrefereed preprints [FvHT23, FvHT24] for sharp Brunn–Minkowski stability. That is a legitimate concern, but I find a more immediate and concrete internal gap: the dimension count in §7.4 is off by two. The slices used to project an n-dimensional problem onto R^2 have dimension n−2, so their Brunn–Minkowski inequality carries exponent 1/(n−2), not 1/n. Because Lemma 7.16 uses q=p/(1+np), for p close to −1/n the resulting q is far below −1/2, the threshold needed by the two-dimensional theorem used immediately afterwards. The overview's statement that q>−1/2 for all p>−1/n matches the corrected formula q=p/(1+(n−2)p), so the detailed lemmas appear to contain a typographical/indexing error rather than a false theorem. However, as written, the proof is not valid for p∈(−1/n,−1/(n+2)]; this is a load-bearing concern because Theorem 1.6 claims all p>−1/n. My recommended verdict is unchanged in the sense that the paper should still be conditional, but the condition is now stronger: the exponent and q formula in §7.4 must be corrected before the main theorem can be accepted. I disagree with the reader's identification of the weakest assumption because the internal dimension mismatch is more immediately fatal, and it does not depend on the status of the external preprints.","tokens_in":71299,"tokens_out":16564,"duration_ms":151668,"concrete_test":"Recompute Lemma 7.17 with the slice dimension m = n−2: replace the hypothesis b^{1/n} ≥ λa^{1/n}+(1−λ)c^{1/n} by b^{1/m} ≥ λa^{1/m}+(1−λ)c^{1/m} and check whether the optimal q is p/(1+pm) rather than p/(1+pn). Concretely, for n=3, p=−0.3, λ=1/2, choose a=1, c=4, u=1, v=8 and any b ≥ (a+c)/2 (true BM in R^1 for these slices); compute the supremum of q satisfying b·M_{λ,−0.3}(1,8) ≥ M_{λ,q}(a,c·8). If it is approximately −0.428 = p/(1+p) rather than −3 = p/(1+3p), then Lemma 7.16 as stated is invalid and the appeal to Theorem 6.1 fails for this p.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The high-dimensional proof of Theorem 1.9 contains an internal dimension error. In §7.4, the slices C_{z,w} are (n−2)-dimensional by Definition 7.4, so Brunn–Minkowski gives |C_{λz+(1−λ)z′}|^{1/(n−2)} ≥ λ|C_{z,w}|^{1/(n−2)} + (1−λ)|C_{z′,w′}|^{1/(n−2)}. The proof of Lemma 7.16 instead invokes BM with exponent 1/n. Lemma 7.17 then defines q = p/(1+pn) and restricts to p∈(−1/(n+2),0). For p∈(−1/n, −1/(n+2)], this q is ≤ −1/2 (e.g., n=3, p=−0.3 gives q=−3), so the integrated two-dimensional functions satisfy only a q-condition with q below the admissible BBL range, and Theorem 6.1, which requires p>−1/2, cannot be invoked. The overview in §3.3 promises a q>−1/2 for every p>−1/n, which would require the corrected value q = p/(1+(n−2)p). As written, the reduction from Proposition 7.11 to Theorem 6.1 does not cover p∈(−1/n, −1/(n+2)], so the proof of Theorem 1.6 is incomplete for an entire interval of p-values even if [FvHT23, FvHT24] are fully correct.","agreement_with_reader":"disagree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":71623,"tokens_out":11266,"duration_ms":107297,"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":[{"comment":"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":"Section 7.4, Lemma 7.16"},{"comment":"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.","section":"Section 7.4, Lemmas 7.16 and 7.17"},{"comment":"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.","section":"Sections 4, 6, 7, and 9"}],"minor_comments":[{"comment":"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.","section":"Section 5, Theorem 5.1"},{"comment":"The name 'Borell-Brascamb-Lieb' appears in headings and in the abstract; it should be 'Borell-Brascamp-Lieb'.","section":"Sections 1.3 and elsewhere"},{"comment":"In the proof after Lemma 7.16 there is a typo: 'there exist s v' should presumably be 'there exists v'.","section":"Section 7.4.1"},{"comment":"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":"Section 7.4, Lemma 7.16"},{"comment":"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.","section":"Section 4, Lemma 4.9"}],"recommendation":"major_revision","confidential_remarks":"The dimension mismatch in Section 7.4 is the main obstacle. It appears to be a local but load-bearing error: the slicing exponent must be 1/(n-2), and the parameter q must be adjusted accordingly. If the authors repair this and supply the missing p=0 and p>0 cases in Lemma 7.16, the paper would be a strong candidate for publication. Given the length and the number of reductions, I could not verify every step; I would recommend a careful second reading of Section 7.4 in particular. The heavy reliance on the authors' own unpublished preprints [FvHT23, FvHT24] is worth editorial attention; if those preprints are not yet accepted, the present manuscript is materially incomplete."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nRead 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.\n\nThe 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.\n\nThe 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.\n\nThere'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.\n\nBottom 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.","headline":"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.","tokens_in":72123,"tokens_out":6021,"would_cite":false,"duration_ms":49921,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["52A40","26B25","39B62"],"pacs":[],"model":"deepseek-v4-flash","headline":"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.","keywords":["Borell-Brascamp-Lieb inequality","Prékopa-Leindler inequality","quantitative stability","Brunn-Minkowski inequality","p-concave functions","L1 approximation","level-set method","transport map"],"falsifier":"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.","tokens_in":71097,"feed_emoji":"📐","tokens_out":9770,"duration_ms":91160,"temperature":0.7,"pith_summary":"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.","feed_headline":"Excess mass forces √δ-close p-concave approximation","feed_subtitle":"The Borell-Brascamp-Lieb family, including Prékopa-Leindler, is now stable at the sharp square-root rate in every dimension.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"It supplies Theorem 1.1, the sharp Brunn-Minkowski stability result that the level-set lemmas invoke as a black box.","marker":"[FvHT23]"},{"why":"It is the companion sharp stability result cited alongside Theorem 1.1 and used for the linear and transport versions of the argument.","marker":"[FvHT24]"},{"why":"It introduced the Borell-Brascamp-Lieb inequality for convex set functions, the object the paper proves stable.","marker":"[Bor75]"},{"why":"It introduced the functional form of the inequality and the $p$-means $M_{\\lambda,p}$, the central objects of the theorem.","marker":"[BL76]"},{"why":"It classifies the equality cases at $p=-1/n$, which is why the main theorem excludes that endpoint.","marker":"[Dub77]"},{"why":"It gave previous non-sharp stability for arbitrary measurable functions, a baseline the new theorem improves to the optimal rate.","marker":"[BFR23]"}],"fun_headline_variants":["Sharp square-root stability proven for Borell-Brascamp-Lieb","Optimal stability for Prekopa-Leindler and Borell-Brascamp-Lieb","Square-root rate settles stability conjecture for Prekopa-Leindler","Excess mass forces square-root-close p-concave approximation","Unified sharp stability for Prekopa-Leindler and Borell-Brascamp-Lieb"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Sharp square-root stability proven for Borell-Brascamp-Lieb","Optimal stability for Prekopa-Leindler and Borell-Brascamp-Lieb","Square-root rate settles stability conjecture for Prekopa-Leindler","Excess mass forces square-root-close p-concave approximation","Unified sharp stability for Prekopa-Leindler and Borell-Brascamp-Lieb"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000944,"raw_usage":{"total_tokens":4037,"prompt_tokens":957,"completion_tokens":3080,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":573,"completion_tokens_details":{"reasoning_tokens":2975}},"tokens_in":573,"tokens_out":3080,"duration_ms":24995,"temperature":1.0,"reasoning_tokens":2975,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T21:26:36.130894+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[],"review_version":1}