{"id":"881a5b25-44ff-4b2d-85fe-c30fc31bcbe7","arxiv_id":"2607.20223","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For every smooth unconditional convex body, the centro-affine Poincaré inequality holds with sharp constant n for all smooth functions satisfying two orthogonality conditions.","lead":"An inequality that controls how much a function on a sphere can change, measured with a shape-dependent geometry, is proved for all smooth symmetric convex bodies with the best possible constant. The result strengthens the case for the log-Minkowski and log-Brunn-Minkowski conjectures in convex geometry.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Equality characterization in Theorem 1.1 rests on two unproved results from the authors' own preprint [HI26]; if either fails, the strictness conclusion collapses.","rationale":"I read the proof as a parity decomposition: F_0 is handled by [HI26, Thm. 8.12], single-odd components by Corollary 2.6, and multi-odd components by Lemma 2.4. The internal steps—Lemma 2.1, Lemma 2.2, Lemma 2.3, the section arguments, and the approximation trick—appear coherent. The deformation and concavity arguments are detailed, and the use of the planar log-Brunn–Minkowski inequality is legitimate because the sections are two-dimensional origin-symmetric bodies. The one place where the proof outsources its central novelty is the equality characterization: Theorem 2.5 is explicitly stated as part of [HI26, Thm. 8.12], and the strict-positivity lemma is quoted from [HI26]. Both are from the authors' own unpublished preprint; [KM22] supports only the inequality, not equality. This is precisely the reader's weakest assumption, so I agree with the conditional assessment. I found no additional internal inconsistency that would require moving to reject or unverdict; supplying [HI26] or proving these two ingredients would likely resolve the dependency.","tokens_in":13108,"tokens_out":12922,"duration_ms":111683,"concrete_test":"Obtain [HI26] and independently verify the two black-boxed ingredients: (a) that Thm. 8.12 really contains the equality characterization 'only when u is constant on Ω_+' and that its proof does not depend on the theorem of the present paper; (b) that Lem. 2.3 proves ⟨P_F x, P_F Dh(x)⟩>0 whenever P_F x≠0 for every C^∞_+ unconditional body. A minimal spot-check is to compute x_iX_i/x_i^2 for a non-ellipsoidal unconditional example, e.g. h=(Σ x_i^4)^{1/4}, on the positive orthant and confirm strict positivity; if Lem. 2.3 is missing or allows zero, the strictness argument in Lemma 2.4 and Theorem 1.1 collapses.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The proof of the central equality statement in Theorem 1.1 imports two results from the authors' unpublished preprint [HI26] without proof. First, Theorem 2.5 is quoted as '[HI26, Thm. 8.12]' and supplies the unconditional-function Poincaré inequality together with the equality characterization 'only when u is constant on Ω_+'. This equality characterization is used in Corollary 2.6 and again in the final equality analysis of Theorem 1.1. Second, [HI26, Lem. 2.3] is invoked in the second proof of Lemma 2.4 to assert strict positivity ⟨b,ξ⟩>0 whenever P_F x≠0; without this strict positivity the argument that equality forces u≡0 does not go through. The manuscript explicitly says 'This is part of [HI26, Thm. 8.12]' and later 'We omit the details' for a related capillary inequality, so these ingredients are not established here. The inequality component alone has independent support from [KM22], but equality cases do not. If [HI26, Thm. 8.12] proves only the inequality, or if [HI26, Lem. 2.3] allows ⟨b,ξ⟩=0 for nonzero b, then the conclusion F≡0 in equality cases of Theorem 1.1 and the equality statements in Corollaries 1.2 and 1.3 do not follow. This is a dependency gap, not an internal inconsistency.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proves a sharp centro-affine Poincaré inequality on S^{n-1} for C∞_+ unconditional convex bodies: if F has zero mean with respect to the cone-volume measure dV_K and is orthogonal to the functions l_i = x_i/h, then n∫F² dV ≤ ∫|∇F|²_g dV, with equality iff F ≡ 0. The proof decomposes F into parity components under coordinate reflections; the fully even component is handled by a result quoted from the authors' preprint [HI26], the components odd in exactly one coordinate by a new orthogonality argument (Corollary 2.6), and the components odd in at least two coordinates by a new section-wise estimate (Lemma 2.4). The paper also derives applications: a local-to-global log-Minkowski inequality with one unconditional body (Corollary 1.2) and a uniqueness result in the supercritical L_p Minkowski problem (Corollary 1.3).","tokens_in":13430,"tokens_out":21931,"duration_ms":195567,"significance":"If the main theorem holds, it establishes the expected spectral gap n for the centro-affine Laplacian on every smooth unconditional convex body, extending earlier results for symmetric bodies and for unconditional functions. The proof is structurally interesting: the parity decomposition is clean, and the reduction to planar sections via Lemma 2.2 and Lemma 2.3 is elegant. The new Lemma 2.4 is a substantial technical contribution. The applications to the log-Minkowski inequality and to supercritical uniqueness are nontrivial and would be of interest to the community. However, the equality characterization and strictness arguments depend on two results quoted without proof from the authors' own unpublished preprint [HI26], so the central claim is not fully self-contained. A related proof in Corollary 1.3 also contains an erroneous factor 1/n, although it appears to be a fixable typo.","major_comments":[{"comment":"Theorem 2.5 is quoted from the authors' preprint [HI26, Thm. 8.12] without proof; the manuscript says only 'This is part of [HI26, Thm. 8.12]'. This result supplies both the inequality and the equality characterization used in Corollary 2.6 and in the F_0 term of Theorem 1.1. The statement in Theorem 2.5 applies to arbitrary u∈C^1(S^{n-1}), while the introduction describes [HI26, Thm. 8.12] as an unconditional-function inequality. Thus the equality characterization 'only when u is constant on Ω_+' is not established in this manuscript. If that characterization fails, the equality statement of Theorem 1.1 (and hence Corollaries 1.2 and 1.3) does not follow. The authors should either include a full proof of Theorem 2.5 and the relevant part of [HI26], or clearly state this as an external result and verify that the hypotheses of [HI26, Thm. 8.12] are satisfied in the needed cases.","section":"Section 2, Theorem 2.5; Corollary 2.6; Theorem 1.1"},{"comment":"The strictness arguments in the second proof of Lemma 2.4 rely on [HI26, Lem. 2.3] to conclude ⟨b,ξ⟩ > 0 whenever b≠0 (or P_F x≠0). Without this positivity lemma, the proof only gives a non-strict inequality, and the equality cases of Theorem 1.1 would collapse. Similarly, Lemma 2.2 uses [HI26, Lem. 2.3] for the strict sign of X_1 X_2. These positivity facts are load-bearing and are not proved in the manuscript. They should be stated as a lemma and proved, or the dependence on [HI26] should be made explicit with a precise statement and the status (published/unpublished) of [HI26] clarified.","section":"Lemma 2.4, second proof; also Lemma 2.2"},{"comment":"The displayed definition of F_i contains an erroneous factor 1/n: F_i = X_i - (V/n)∑_j (A^{-1})_{ij} l_j yields ∫ F_i l_k dV = V(1 - 1/n)δ_{ik} ≠ 0, so Theorem 1.1 cannot be applied. The subsequent algebra — in particular ∑_i ∫ F_i² dV = ∫|X|² dV - V² tr(A^{-1}) — corresponds to the definition F_i = X_i - V∑_j (A^{-1})_{ij} l_j, without the factor 1/n. Please correct this typo; the proof is sound after this correction.","section":"Corollary 1.3, proof"}],"minor_comments":[{"comment":"The paper states a half-space capillary inequality and then says 'We omit the details'. Since this result is not used in the main theorem and is presented without proof, it should be framed as a remark or conjecture rather than as a proved assertion.","section":"Section 1, capillary example"},{"comment":"The one-line proof is only a reference to [HI26, Thm. 8.12]. If the journal permits citing a preprint, the exact statement from [HI26] should be quoted and the hypotheses checked, especially because the statement here is broader than the unconditional-function description in the introduction.","section":"Section 2, Theorem 2.5"},{"comment":"The term dS_{D0} in the final integral is not defined. Please define the surface measure used, or use dV_{D0} if that is intended.","section":"Lemma 2.3"},{"comment":"The assertion that w_ξ(a) depends smoothly on (ξ,a) is stated without details. Please spell out the regularity of the minimizer in (2.4) and the dependence on the section parameter ξ.","section":"First proof of Lemma 2.4, after (2.8)"},{"comment":"The paper uses notation and conventions from [HI26] (connections, Laplacian, dV_K, etc.) that are not fully defined in the present text. Adding a short list of conventions would improve readability.","section":"Throughout"},{"comment":"Several references are to unpublished preprints ([HI26], [CHI26], [Iff26], [Du25]). If these are publicly available, please include arXiv identifiers and submission dates so that the reader can verify the cited results.","section":"References"}],"recommendation":"major_revision","confidential_remarks":"The manuscript depends very heavily on the authors' own preprint [HI26], both for the equality characterization in Theorem 2.5 and for the positivity lemma used in the strictness arguments. As editor, you may wish to verify that [HI26] is publicly available and that the cited theorems actually contain the needed statements in the needed breadth. The factor-1/n typo in Corollary 1.3 is straightforward to fix but should be corrected before publication. The central idea appears sound, but the paper is not yet self-contained enough for the strong equality claims it makes."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Here is my take. The main theorem is real: for every smooth unconditional convex body, the centro-affine Laplacian has spectral gap n on the subspace orthogonal to 1 and the coordinate functions, with equality only for zero. The parity decomposition is the right trick, and the new Lemma 2.4 is the real work. Both proofs of Lemma 2.4 are coherent; the second one, using the planar log-Brunn–Minkowski inequality to get concavity of Vol(K_t), is genuinely elegant. Lemma 2.2 is a clean integration by parts. I found no algebraic errors.\n\nThe soft spot is dependency, not correctness. The equality characterization in Theorem 1.1 imports Theorem 2.5 (quoted from [HI26, Thm. 8.12]) and Lemma 2.3 from [HI26]. The proof of Theorem 2.5 in this manuscript is four lines and essentially says 'this is part of [HI26]'. Corollary 2.6 and the strictness arguments in Lemma 2.4 rely on the equality characterization and on strict positivity <b,ξ> > 0 from [HI26, Lem. 2.3]. If either of those results fails, the equality cases in Theorem 1.1 and Corollaries 1.2 and 1.3 collapse. The inequality component has independent published support in [KM22], so the gap is specifically the equality part. That is a real gap for a standalone paper, though it looks like a dependency, not an internal contradiction.\n\nThere is also a small aside where the authors say 'we omit the details' for a capillary inequality; that is not load-bearing.\n\nMy verdict: the result is likely correct, but the paper is not self-contained in an essential way. It deserves a serious referee, but the referee needs access to [HI26] or the authors should include the missing statements/proofs. For publication, I would ask them to either prove or at least state fully the imported results. If [HI26] is available and correct, this is a strong contribution.","headline":"Sharp centro-affine Poincaré inequality for unconditional bodies, with an elegant parity argument; the main caveat is heavy reliance on the authors' own unpublished preprint for the equality cases.","tokens_in":13930,"tokens_out":3548,"would_cite":true,"duration_ms":29839,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["52A40","35P15","53A15"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves that the centro-affine Poincaré inequality holds with the sharp constant n on every smooth unconditional convex body, with equality only for the zero function.","keywords":["centro-affine Poincaré inequality","unconditional convex body","spectral gap","cone-volume measure","even log-Minkowski inequality","L_p Minkowski uniqueness","parity decomposition","centro-affine Laplacian"],"falsifier":"Find a smooth unconditional convex body and a nonzero smooth function F satisfying both orthogonality conditions with n∫F² dV_K = ∫|∇F|² dV_K; this would disprove the equality characterization in Theorem 1.1. A concrete place to look is the equality case of the companion unconditional-function theorem: if any nonconstant unconditional function attains equality there, the main theorem's equality claim collapses.","tokens_in":12987,"feed_emoji":"📐","tokens_out":6479,"duration_ms":60731,"temperature":0.7,"pith_summary":"This paper proves a sharp spectral-gap estimate for the centro-affine Laplacian on every smooth unconditional convex body: any smooth function orthogonal to constants and to the n coordinate functions l_i = x_i/h satisfies n∫F² dV ≤ ∫|∇F|² dV, with equality only for F ≡ 0. Previously this was known for unconditional functions; the paper removes the symmetry restriction on F by decomposing F into 2^n parity components and handling each type of component. The one-coordinate-odd components are reduced to the unconditional case by adding a suitable multiple of l_i, while the two-coordinate-odd components are treated by slicing the body into two-dimensional sections and applying planar inequalities fiberwise. What makes this worth caring about is that the strict spectral gap n directly yields the even log-Minkowski inequality when one body is unconditional, and the uniqueness of solutions to the L_p Minkowski problem in the supercritical range p ∈ [-n-1, -n) for n ≥ 3.","feed_headline":"Sharp spectral gap n proven for all unconditional convex bodies","feed_subtitle":"Extends the even log-Minkowski inequality and supercritical uniqueness to every such body.","key_machinery":"The central device is the parity decomposition of F into components even or odd under each coordinate reflection. For the component odd in exactly one coordinate i, the paper adds a carefully chosen multiple of the eigenfunction l_i = x_i/h (which satisfies ∆l_i = -(n-1)l_i) to make it mean-zero on the positive orthant, then applies the known unconditional-function Poincaré inequality; the orthogonality condition ∫ F l_i dV_K = 0 ensures the added term does not change the quadratic form. For components odd in two coordinates i and j, the paper slices K into two-dimensional sections parallel to the (i,j)-plane. On each section the induced speed function is odd in both coordinate reflections,","core_discovery":"The paper establishes Theorem 1.1: for a C^∞_+ unconditional convex body K in R^n, equipped with the centro-affine metric g = (1/h)(∇̄²h + hḡ) and cone-volume measure dV_K, every smooth function F on the sphere with ∫ F dV_K = 0 and ∫ F (x_i/h) dV_K = 0 for each i satisfies n∫ F² dV_K ≤ ∫ |∇F|²_g dV_K, with equality if and only if F ≡ 0. This is the sharp Poincaré constant n on the subspace orthogonal to the constant function and the n coordinate functions l_i = x_i/h. The proof is structural rather than analytic: it decomposes F into 2^n pieces by parity with respect to the coordinate reflections, proves the inequality separately for pieces whose parity pattern has zero, one, and at least t","pith_inferences":["The slicing proof suggests an inductive strategy for the same inequality on arbitrary convex bodies: if every two-dimensional coordinate section satisfied the planar inequality uniformly, Fubini would give the n-dimensional result; this may point toward a proof of the full centro-affine Poincaré inequality by dimension induction.","The equality characterization in the companion unconditional-function theorem is load-bearing; if it were replaced by a non-sharp version, the strictness in Theorem 1.1 and the uniqueness in Corollary 1.3 would fail, but the inequality itself might still hold.","A quantitative version of the strict gap (λ > n) could yield stability estimates for the even log-Minkowski inequality for unconditional bodies, measuring how close a body is to being a homothetic copy in terms of the deficit."],"forward_implications":["The even log-Minkowski inequality holds for every origin-symmetric convex body L when K is a C^∞_+ unconditional convex body, with equality only if L = cK (Corollary 1.2).","For n ≥ 3 and -n-1 ≤ p < -n, the only smooth unconditional convex bodies with h^{1-p}_K = 1 are unit balls, settling uniqueness in that supercritical range (Corollary 1.3).","The first nonzero even eigenvalue of the centro-affine Laplacian on an unconditional body is strictly greater than n, because equality in Theorem 1.1 forces the function to vanish.","The strict inequality with equality characterization upgrades the earlier inequality-only results for unconditional bodies to a full spectral-gap statement with no extremal functions."],"fun_headline_variants":["Sharp spectral gap n for smooth unconditional convex bodies","Unconditional smooth bodies get sharp Poincaré constant n","Poincaré inequality: constant n sharp for C∞+ unconditional bodies","Centro-affine spectral gap n proven for smooth unconditional sets"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The argument relies on a companion paper's equality characterization for unconditional functions and a strict-positivity lemma; if either of those is wrong, the strictness and equality conclusions of the main theorem do not follow.","fun_headline_variants_meta":{"raw":{"variants":["Sharp spectral gap n for smooth unconditional convex bodies","Unconditional smooth bodies get sharp Poincaré constant n","Poincaré inequality: constant n sharp for C∞+ unconditional bodies","Centro-affine spectral gap n proven for smooth unconditional sets"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000729,"raw_usage":{"total_tokens":3022,"prompt_tokens":587,"completion_tokens":2435,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":331,"completion_tokens_details":{"reasoning_tokens":2366}},"tokens_in":331,"tokens_out":2435,"duration_ms":21500,"temperature":1.0,"reasoning_tokens":2366,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-01T10:26:52.469304+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Find a smooth unconditional convex body and a nonzero smooth function F satisfying both orthogonality conditions with n∫F² dV_K = ∫|∇F|² dV_K; this would disprove the equality characterization in Theorem 1.1. A concrete place to look is the equality case of the companion unconditional-function theorem: if any nonconstant unconditional function attains equality there, the main theorem's equality claim collapses.","supporting_citations":[],"review_version":1}