{"id":"2725d257-9e30-4a77-b3c7-9afeaff573e7","arxiv_id":"2506.16941","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Weighted marginals of even concave functions are concave after a 1/(β+n) power for rotationally invariant log-concave measures, unifying the functional dimensional Brunn-Minkowski and B-inequalities under a new 'hereditary convexity' condition.","lead":"This paper proves new concavity principles for weighted marginals of even concave functions under rotationally invariant log-concave measures. It yields a functional version of the dimensional Brunn-Minkowski inequality and a Prékopa-type theorem containing the B-inequality.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Proposition 9 imports inequality (63) from [25, Theorem 4] for arbitrary even ν log-concave w.r.t. µ with bounded support and arbitrary Neumann data; if [25] does not cover this generality, estimate (74) in Theorem 10's Step 1 has no basis.","rationale":"The reader's weakest assumption correctly identifies hereditary convexity, inequality (59), as the load-bearing external input. My stress-test confirms that the internal derivation of Proposition 9 is algebraically sound once inequality (63) is granted, so the decisive question is whether [25, Theorem 4] has the full scope demanded by Definition 8(ii): arbitrary even ν that is log-concave w.r.t. µ, possibly supported on a bounded convex set with smooth boundary, and functions with arbitrary Neumann data. This scope is exactly what Step 1 needs because u solves (43)–(44) with nonzero Neumann data. The reader and I agree on this central concern; I add only the boundary-condition precision and a way to settle it. Since the concern is real but not yet demonstrated to be fatal, the reader's CONDITIONAL verdict remains appropriate.","tokens_in":31575,"tokens_out":15355,"duration_ms":165995,"concrete_test":"Obtain the precise statement of [25, Theorem 4]. Check whether it allows (i) ν with density e^{-Ψ}dµ for arbitrary even convex Ψ, (ii) supp ν a bounded C²-smooth convex set, and (iii) arbitrary Neumann boundary data for v. If any of these fails, recompute Proposition 9 keeping the boundary terms from the Bochner–Reilly identity (21): either prove they are nonnegative for v = u + (λ/2)|x|² under the evenness assumptions, or construct an explicit n=1 example (µ = standard Gaussian, ν = Gaussian restricted to [-1,1], even u with u'(1) ≠ 0) where (63) fails. Either outcome settles whether estimate (74) is available.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The proof of Theorem 1 rests on the hereditary-convexity estimate (59) in Definition 8; in Step 1 of Theorem 10 this is the only estimate controlling the u-terms, via (74). For the rotationally invariant class, Proposition 9 reduces (59) to inequality (63), stated without proof and attributed to [25, Theorem 4], applied to v = u + (λ/2)|x|². The application is made for every even probability ν of the form dν = e^{-Ψ}dµ with Ψ even convex, on a possibly bounded smooth convex support U, and for u with arbitrary Neumann data on ∂U. These boundary conditions are not cosmetic: the elliptic PDE used in Step 1 has nonzero Neumann data (44), so the Bochner–Reilly identity (21) contains boundary terms. If [25, Theorem 4] is proved only on R^n or only under zero Neumann data, those boundary terms are missing from (63) and the passage from (63) to (59) is unjustified. Every algebraic step after (63) is correct; the load-bearing unknown is the exact scope of the cited theorem.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper develops a local L2/Bochner method for weighted marginal functions of the form φ(t)=(∫_{Ω_t} Φ(t,x)^β dμ(x))^{1/(β+n)}. It introduces a spectral condition called 'hereditary convexity' (Definition 8) and shows, in Theorem 10, that this condition suffices for a functional dimensional Brunn–Minkowski inequality; Proposition 9 then asserts that rotationally invariant measures with density e^{-w(|x|)}, where t↦w(e^t) is increasing and convex, satisfy the condition. This yields Theorem 1, the main concavity principle, and Corollary 2 for weighted Brunn–Minkowski inequalities. A second application gives Theorem 4, a log-concavity preservation principle for weighted marginals with a Hessian lower bound, which contains a functional B-inequality for rotationally invariant measures. Theorem 3 derives a weighted Poincaré–Brascamp–Lieb inequality from Theorem 1. The paper also discusses negative exponents, questions on torsional rigidity, and reprints Brascamp–Lieb's proof of the Gaussian B-theorem in an appendix.","tokens_in":31797,"tokens_out":16336,"duration_ms":176245,"significance":"If all claims hold, the paper is a substantial contribution: it upgrades the geometric dimensional Brunn–Minkowski inequality for rotationally invariant measures of [27,25] to a functional version, and it provides a unified hereditary-convexity framework that also yields the B-inequality. The second-derivative formula in Proposition 7 and the approximation chain in the proof of Theorem 10 are technically valuable, and the authors are transparent about the points where the proof relies on the earlier spectral result [25, Theorem 4]. The main concern is that this cited result is not stated in sufficient detail for the reader to verify that it covers the bounded-support, arbitrary-Neumann-data case required in the proof of Proposition 9; this issue is load-bearing for Theorem 1. The paper also contains a display error in Theorem 3 and an invalid illustrative counterexample in the introduction.","major_comments":[{"comment":"The proof of Proposition 9 applies [25, Theorem 4] to v = u + (λ/2)|x|² and concludes (63), but the cited theorem is not stated, and no argument is given for the case allowed in Definition 8(ii) where the measure ν has bounded smooth support U and the function u has arbitrary Neumann data on ∂U. This is exactly the case needed in Step 1 of the proof of Theorem 10 via inequality (74), since the measure ν there is supported on the bounded set Ω0 and u satisfies the nonzero Neumann condition (44); the boundary terms in the weighted Reilly formula (21) are then generally present. Please state Theorem 4 of [25] and either verify that its hypotheses cover the bounded-support Neumann-data situation or supply a proof of (63) in that situation. Without this, the passage from (63) to the hereditary-convexity inequality (59), and hence the central estimate (74), is not justified as written.","section":"Section 4.1 (Proposition 9, inequality (63)) and Section 4.2 (Step 1, inequality (74))"},{"comment":"The displayed inequality (12) contains ⟨(-∇²Φ)∇g,∇g⟩ in the first integral, but the proof establishes ⟨(-∇²Φ)^{-1}∇g,∇g⟩. The inverse is what follows from the Cauchy–Schwarz step in the construction of Φ_ε in (99)–(100) and is what appears in (103). As printed, Theorem 3 states a different and unproved inequality; please correct (12) by adding the inverse in the first integrand.","section":"Section 6, Theorem 3, Eq. (12) and proof, Eq. (103)"}],"minor_comments":[{"comment":"The proposed disproof of [60, Conjecture 6.2] using translated indicator functions with M→∞ is not valid as stated for a finite measure μ: by dominated convergence, both ∫ f_M dμ and ∫ g_M dμ tend to zero, so the inequality in question would hold in the limit. The example should be replaced or the argument amended.","section":"Section 1.2"},{"comment":"The displayed implication 'μ verifies the dimensional Brunn–Minkowski conjecture ⇒ μ is even hereditarily convex' appears to have the direction opposite to the surrounding discussion and to Theorem 10, which derives the functional Brunn–Minkowski statement from hereditary convexity. Please clarify the intended logical direction.","section":"Section 4.1, display (62)"},{"comment":"In the approximation argument, the boundary value of Ψ_{1/k} on ∂Ω_{k,m} is η+1/m, whereas Step 1 is formulated for a boundary value η. This is harmless after rescaling the constant, but it should be said explicitly.","section":"Section 4.2, Step 2"},{"comment":"There are several typos that should be corrected in revision: 'Luster nik' (§1.1), 'heriditarily' (Definition 8 and §4.1), 'indespensable' (§2.1), and 'assets' (Remark 14).","section":"Throughout"},{"comment":"Formula (93) is obtained by a limiting argument from Proposition 7 with Φ=(1-β^{-1}V)_+ and β→∞; a short direct derivation, or a precise statement of the approximation used for the unbounded domain, would improve readability and rigor.","section":"Section 5, proof of Theorem 4"}],"recommendation":"major_revision","confidential_remarks":"The decisive point is the scope of [25, Theorem 4] in Proposition 9. If the authors can confirm that this theorem covers bounded supports and arbitrary Neumann data — or can supply a proof of (63) in that setting — the central argument of Theorem 1 will be sound and the paper is likely acceptable after local corrections. If not, the main theorem does not follow as written. The Theorem 3 display error and the invalid [60] counterexample should also be fixed; both are local but should not appear in the final version."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Core message: this is a genuine advance. Theorem 1 gives a functional version of the dimensional Brunn-Minkowski inequality for rotationally invariant log-concave measures (and some non-log-concave ones), covering convex sections that are not product domains. That closes the gap left by Nguyen for β > 0. The hereditary convexity framework is a good organizing idea, and the mixed geometric-functional L2 second-derivative formula in Proposition 7 is new and likely reusable. Theorem 4's log-concavity principle and the Poincaré-type Theorem 3 are solid extras, and the Brascamp-Lieb appendix is a nice service to the reader.\n\nThe proof is long but readable. The approximation steps are spelled out, and the algebra in Proposition 7 checks out. The main line—hereditary convexity propagated through a second-derivative formula, with boundary terms controlled by the level-set assumption on Φ—is coherent. As far as I can tell, the central theorem stands.\n\nThree soft spots, none fatal. First, the counterexample to [60, Conjecture 6.2] in §1.2 is not valid as stated: for a finite measure, μ(B_n^2 + M e_1) → 0 by dominated convergence, so the proposed inequality is not contradicted in the limit. Replace it with a working example or soften the claim. Second, Proposition 9 imports inequality (63) from [25, Theorem 4] for arbitrary even ν with bounded support and arbitrary Neumann data. This is load-bearing: the Bochner–Reilly identity has boundary terms, and estimate (74) needs (59) in exactly that generality. I have no reason to think [25] fails there—one of the authors proved both—but the paper should state precisely what theorem is being imported and under what boundary conditions, rather than leaving the reader to guess. Third, the final paragraph of §7.2 claims that Corollary 17 extends to all β > 0 by inspection of proofs. That is an unproved assertion; supply the argument or remove the sentence.\n\nThese are revision items, not reasons to doubt the main theorems. The citation pattern is honest, and the reliance on prior work is explicit. The paper is not fully self-contained, but that is normal for this line of research.\n\nWho it is for: anyone working on Brunn–Minkowski inequalities, log-concave measures, or L2/Bochner methods. It deserves a serious referee. Fix the counterexample, make the imported spectral estimate precise, and settle the β > 0 claim, and I would be comfortable accepting.","headline":"A genuine functional Brunn-Minkowski advance for rotationally invariant measures, with a few fixable rough edges that should be addressed before publication.","tokens_in":32344,"tokens_out":2540,"would_cite":true,"duration_ms":25558,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["52A40","52A20","28C20","60D05","47F10"],"pacs":[],"model":"deepseek-v4-flash","headline":"For rotationally invariant measures, weighted marginals of concave functions with symmetric sections are concave after the $1/(\\beta+n)$-power, for every $\\beta>0$.","keywords":["Brunn–Minkowski inequality","B-inequality","weighted marginals","log-concave measures","rotationally invariant measures","hereditary convexity","Borell–Brascamp–Lieb inequalities","symmetric convex sets"],"falsifier":"Search over admissible rotationally invariant weights $w$, even log-concave measures $\\nu$ on smooth symmetric supports, and even functions $u$ with compatible Neumann data: a single instance in which the left side of (59) is smaller than the right side for an allowed weight would contradict Proposition 9 and remove the support for Theorem 1. A more direct but harder test is to find any convex $\\Omega$ with symmetric sections and concave even $\\Phi$ for which the function $\\varphi$ in (8) fails to be concave.","tokens_in":31365,"feed_emoji":"📐","tokens_out":12985,"duration_ms":114675,"temperature":0.7,"pith_summary":"The paper proves a concavity principle for weighted marginals under rotational symmetry. If $d\\mu(x)=e^{-w(|x|)}dx$ with $w$ increasing and $t\\mapsto w(e^t)$ convex, then for every concave $\\Phi$ with even sections on a convex set $\\Omega$ with symmetric sections and every $\\beta>0$, the function $\\varphi(t)=(\\int_{\\Omega_t}\\Phi(t,x)^\\beta\\,d\\mu(x))^{1/(\\beta+n)}$ is concave on its support. This is the functional form of the dimensional Brunn–Minkowski inequality for such measures, and it applies even when $\\mu$ itself is not log-concave, for instance for weights like $(1+|x|^\\alpha)^{-\\beta}$. The same machinery yields a Prékopa-type log-concavity theorem for weighted marginals that contains the B-inequality for rotationally invariant measures, as well as a weighted Poincaré–Brascamp–Lieb inequality.","feed_headline":"Weighted marginals stay concave under rotationally invariant measures","feed_subtitle":"For every β>0, the (β+n)-root marginal is concave: a functional Brunn-Minkowski principle for symmetric weights.","key_machinery":"The load-bearing object is the hereditary convexity condition (Definition 8): an even measure $\\mu=e^{-W}dx$ satisfies inequality (59), namely $\\int(\\|\\nabla^2 u\\|_{\\mathrm{HS}}^2+\\langle\\nabla^2 W\\,\\nabla u,\\nabla u\\rangle)\\,d\\nu \\ge (\\int L_\\mu u\\,d\\nu)^2/\\int L_\\mu(|x|^2/2)\\,d\\nu$ for every even probability measure $\\nu$ log-concave with respect to $\\mu$ and every even smooth $u$ with Neumann data. That condition is exactly the curvature lower bound needed to make the second derivative of the marginal function nonpositive. The second-derivative formula itself (Proposition 7) is derived by a mixed geometric-functional $L^2$ computation: support-function perturbations of the convex sections (Lemmas 5 and 6) are combined with the Bochner–Reilly identity and an elliptic PDE solved by a function $u$, following the line of Brascamp–Lieb and Nguyen but extended to non-product domains. Proposition 9 proves hereditary convexity for rotationally invariant $\\mu$ using a weighted Poincaré inequality for even functions imported from [25, Theorem 4].","core_discovery":"Restated on the paper's own terms, the central result is: for every increasing $w$ with $t\\mapsto w(e^t)$ convex, the measure $d\\mu(x)=e^{-w(|x|)}dx$ on $\\mathbb{R}^n$, every convex $\\Omega\\subset\\mathbb{R}^{n+1}$ whose sections $\\Omega_t$ are symmetric, and every concave $\\Phi:\\Omega\\to\\mathbb{R}_+$ that is even in $x$ on each section, the marginal function $\\varphi(t)=(\\int_{\\Omega_t}\\Phi(t,x)^\\beta\\,d\\mu(x))^{1/(\\beta+n)}$ is concave on its support for every $\\beta>0$, whenever the integral converges. The paper also proves that for $\\kappa\\in[0,1]$, if $V\\in C^2$ has even sections and satisfies the Hessian lower bound (14), then $\\alpha(t)=\\int_{\\mathbb{R}^n} e^{-V(t,x)}\\,d\\mu(x)$ is log-concave; the case $\\kappa=1$ contains the functional B-theorem, and hence the B-inequality, for rotationally invariant measures. These statements are established by approximation from smooth cases and by showing that rotationally invariant measures of the stated class satisfy a spectral property the paper calls hereditary convexity.","pith_inferences":["If the hereditary convexity question posed for all even log-concave measures has an affirmative answer, the same proof structure would deliver both the B-conjecture and the dimensional Brunn–Minkowski conjecture for general even measures; rotational invariance is the sufficient case established here.","The second-derivative formula (45) is a general calculus for weighted marginals that is likely to transfer to other functionals; the paper's question about weighted torsional rigidity is the natural next target.","For non-rotationally invariant even log-concave measures, inequality (59) can be probed directly on quadratic test functions $u$; the first even counterexample, if it exists, would show exactly where symmetry assumptions are needed.","The proof uses that $\\Phi^\\beta d\\mu$ is more log-concave than $\\mu$, so the negative-exponent regime of Question 19 will require new spectral inequalities rather than a simple adaptation of this argument."],"forward_implications":["Corollary 2: for each $\\beta>0$, the measure $\\nu_\\beta=\\Phi^\\beta d\\mu$ on a symmetric convex set $C$ satisfies $\\nu_\\beta(\\lambda K+(1-\\lambda)L)^{1/(\\beta+n)}\\ge \\lambda\\nu_\\beta(K)^{1/(\\beta+n)}+(1-\\lambda)\\nu_\\beta(L)^{1/(\\beta+n)}$ for all symmetric convex $K,L$.","Theorem 4 with $\\kappa=1$ yields the functional B-theorem, the log-concavity of $t\\mapsto\\int e^{-V(e^t x)-W(x)}\\,dx$, and hence the B-inequality for rotationally invariant measures; Remark 13 notes that this provides a new proof of the B-theorem for such measures.","Theorem 3 gives a weighted Poincaré–Brascamp–Lieb inequality (12) for even functions under the same class of rotationally invariant weights.","The approximation argument repairs a gap in earlier local proofs of dimensional Prékopa theorems in the range $\\beta>0$, where only product-set domains were handled, and extends the conclusion to general convex domains with symmetric sections."],"supporting_citations":[{"why":"Supplies the weighted Poincaré inequality for even functions on rotationally invariant measures that proves hereditary convexity in Proposition 9.","marker":"[25]"},{"why":"Provides the Bochner–Reilly boundary formula and support-function perturbation calculus used to compute the second derivative of marginals.","marker":"[41]"},{"why":"Gives the local $L^2$/elliptic-PDE proof of the dimensional Prékopa theorem that the paper extends from product domains to general convex domains.","marker":"[55]"},{"why":"Establishes the Brascamp–Lieb variance inequality and the functional Prékopa–Leindler theory whose weighted refinement is at issue.","marker":"[16]"},{"why":"Classifies convex measures and supplies the Borell–Brascamp–Lieb concavity principles that set the $\\beta>0$ concavity regime.","marker":"[8]"},{"why":"Proves the dimensional Brunn–Minkowski inequality in Gauss space and contributes the spectral method adapted here.","marker":"[27]"},{"why":"Shows the equivalence between the B-inequality and its functional form, which frames the $\\kappa=1$ case of Theorem 4.","marker":"[24]"},{"why":"Prékopa's theorem is the log-concave baseline that Theorem 4 extends under evenness and rotational symmetry.","marker":"[57]"},{"why":"Provides the universal dimensional Brunn–Minkowski bound used to derive concavity for arbitrary $\\beta>0$ in Corollary 17.","marker":"[47]"}],"fun_headline_variants":["Functional Brunn-Minkowski for rotationally invariant measures","Prékopa-type concavity with symmetric weights","B-inequality extended to rotationally invariant measures","Concavity principles for weighted marginals under symmetry","Weighted marginals concave for symmetric measures"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof rests on a single spectral bound: for every symmetric log-concave perturbation of the measure and every symmetric test function, a curvature integral always dominates the square of an average divided by a dimension factor; for rotationally invariant weights this bound is imported from an earlier theorem, and if it failed for an admissible weight, the concavity conclusion would no longer follow.","fun_headline_variants_meta":{"raw":{"variants":["Functional Brunn-Minkowski for rotationally invariant measures","Prékopa-type concavity with symmetric weights","B-inequality extended to rotationally invariant measures","Concavity principles for weighted marginals under symmetry","Weighted marginals concave for symmetric measures"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001301,"raw_usage":{"total_tokens":5277,"prompt_tokens":884,"completion_tokens":4393,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":500,"completion_tokens_details":{"reasoning_tokens":4318}},"tokens_in":500,"tokens_out":4393,"duration_ms":32524,"temperature":1.0,"reasoning_tokens":4318,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T19:17:36.776758+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Search over admissible rotationally invariant weights $w$, even log-concave measures $\\nu$ on smooth symmetric supports, and even functions $u$ with compatible Neumann data: a single instance in which the left side of (59) is smaller than the right side for an allowed weight would contradict Proposition 9 and remove the support for Theorem 1. A more direct but harder test is to find any convex $\\Omega$ with symmetric sections and concave even $\\Phi$ for which the function $\\varphi$ in (8) fails to be concave.","supporting_citations":[{"cited_title":"Improved log-concavity for rotationally invariant measures of sym- metric convex sets","cited_arxiv_id":null,"evidence_quote":"Supplies the weighted Poincaré inequality for even functions on rotationally invariant measures that proves hereditary convexity in Proposition 9."},{"cited_title":"Kolesnikov and Emanuel Milman","cited_arxiv_id":null,"evidence_quote":"Provides the Bochner–Reilly boundary formula and support-function perturbation calculus used to compute the second derivative of marginals."},{"cited_title":"A local proof of the dimensional Pr´ ekopa’s theorem","cited_arxiv_id":null,"evidence_quote":"Gives the local $L^2$/elliptic-PDE proof of the dimensional Prékopa theorem that the paper extends from product domains to general convex domains."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Establishes the Brascamp–Lieb variance inequality and the functional Prékopa–Leindler theory whose weighted refinement is at issue."},{"cited_title":"Convex set functions in d-space","cited_arxiv_id":null,"evidence_quote":"Classifies convex measures and supplies the Borell–Brascamp–Lieb concavity principles that set the $\\beta>0$ concavity regime."},{"cited_title":"The dimensional Brunn-Minkowski inequality in Gauss space","cited_arxiv_id":null,"evidence_quote":"Proves the dimensional Brunn–Minkowski inequality in Gauss space and contributes the spectral method adapted here."},{"cited_title":"Several results regarding the (B)-conjecture","cited_arxiv_id":null,"evidence_quote":"Shows the equivalence between the B-inequality and its functional form, which frames the $\\kappa=1$ case of Theorem 4."},{"cited_title":"On logarithmic concave measures and functions.Acta Sci","cited_arxiv_id":null,"evidence_quote":"Prékopa's theorem is the log-concave baseline that Theorem 4 extends under evenness and rotational symmetry."},{"cited_title":"Livshyts","cited_arxiv_id":null,"evidence_quote":"Provides the universal dimensional Brunn–Minkowski bound used to derive concavity for arbitrary $\\beta>0$ in Corollary 17."}],"review_version":2}