{"id":"ad52f096-110a-4f0a-a484-af4872ca60d4","arxiv_id":"2502.05430","paper_version":1,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":8.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"A non-zero even measure on the sphere is the cone-volume measure of an origin-symmetric convex body exactly when it satisfies the subspace concentration condition.","lead":"This paper gives the exact condition under which a measure on the unit sphere is the cone-volume measure of a symmetric convex body: the subspace concentration condition. It resolves the even logarithmic Minkowski problem, a central existence question in convex geometry and in the geometry of finite-dimensional Banach spaces.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified; the equality-case argument in Theorem 5.2, the most delicate step, is internally sound.","rationale":"The reader correctly identified the equality case of Theorem 5.2 as the most delicate step, and I focused my stress test there. Tracing the argument in detail: the equality P_{ξ⊥}B(ρ_{K~}(u)u)=B(o) for all u gives equality of the slice volumes at the radial boundary of P_ξK; Brunn–Minkowski concavity of the slice-volume function then forces all slices to have the same volume; equality in Brunn–Minkowski for every pair of slices forces them to be translates; and convexity forces the translation locus to be affine in the base point x, hence the body is a Minkowski sum of a slice and an m-dimensional convex set. This is sound. I also checked the sufficiency side: Lemma 6.2's partition argument and Abel summation are correct, and Theorem 6.3's use of John's ellipsoid is standard. Lemma 7.2's non-orthogonal coarea handling was a possible hidden Jacobian issue, but direct computation with a non-orthogonal pair confirms the displayed formula, with the factor r entering through V(D′) rather than through an omitted Jacobian. The n=1 base case in Theorem 7.3 is technically not covered by Theorem 6.3, but it is elementary and does not affect the main theorem. Overall, the central claim survives scrutiny; the ACCEPT verdict requires no change.","tokens_in":18655,"tokens_out":20315,"duration_ms":206567,"concrete_test":"Analytically verify the equality case of Theorem 5.2: for an origin-symmetric convex K with all slices parallel to ξ⊥ of equal (n−m)-volume, prove that the unique translate vector t(x) defined by (x+ξ⊥)∩K = (ξ⊥∩K)+t(x) is affine in x using the inclusion λK_{x1}+(1−λ)K_{x2}⊂K_{λx1+(1−λ)x2} and Brunn–Minkowski equality; if this affine identity fails for some convex K, the cylinder decomposition K=(ξ⊥∩K)+C and the existence of ξ′ in Theorem 5.2 are not established.","verdict_should_be":"UNCHANGED","load_bearing_attack":"I reviewed the proof of Theorem 1.1 with particular attention to the necessity direction and to the compactness argument for sufficiency. The most delicate point is the equality case in Theorem 5.2: from P_{ξ⊥}B(ρ_{K~}(u)u)=B(o) for all u∈S^{m-1}, the concavity of the slice-volume function x↦V_{n−m}(K∩(x+ξ⊥)) implies constant fiber volumes on P_ξK; Brunn–Minkowski equality then forces each fiber to be a translate of ξ⊥∩K, and convexity forces the translation locus to be an origin-symmetric convex C⊂ξ, giving K=(ξ⊥∩K)+C and a complementary subspace ξ′. Each implication is justified by standard Brunn–Minkowski equality conditions; no hidden assumption or circular step appears. The compactness proof in Theorem 6.3 via John's ellipsoid and Lemma 6.2 also checks out: the cross-polytope reduction, the Abel summation estimate, and the contradiction from unboundedness of Φ_μ are valid. The only cosmetic issue is that Theorem 7.3 states n≥1 while Theorem 6.3 is stated for n≥2; the n=1 case is trivial and does not affect the central claim. No load-bearing gap found.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper solves the even logarithmic Minkowski problem: Theorem 1.1 states that a non-zero finite even Borel measure on S^{n-1} is the cone-volume measure of an origin-symmetric convex body in R^n if and only if it satisfies the subspace concentration condition. The sufficiency is proved variationally: one minimizes the logarithmic functional Φ_μ(K)=∫ log h_K dμ under a fixed volume constraint, proves existence of a minimizer by a compactness argument (Theorem 6.3) that uses John's ellipsoid, a cross-polytope estimate (Lemma 6.2), and Blaschke selection, and then shows (Lemma 4.1) that any minimizer has cone-volume measure μ. Equality cases in the subspace concentration inequality are handled by induction on dimension, using Lemmas 7.1 and 7.2 to glue together bodies in complementary subspaces. The necessity of the subspace concentration condition is proved in Theorem 5.2 by a subspace symmetrization argument and an analysis of equality in the Brunn-Minkowski inequality.","tokens_in":18884,"tokens_out":10010,"duration_ms":93655,"significance":"If the result is correct, Theorem 1.1 completely resolves the existence part of the even logarithmic Minkowski problem, a central open problem in convex geometry. The subspace concentration condition is a clean, intrinsic condition that is both necessary and sufficient, so the theorem provides a definitive answer to a question that had only partial solutions for polytopes and smooth bodies. The proof is self-contained modulo standard tools (Brunn-Minkowski, John's theorem, Blaschke selection, Aleksandrov's lemma) and is notable for treating measure data directly rather than by approximation from the smooth case, which the authors explicitly explain is not possible. The equality-case analysis that characterizes the subspace concentration condition (not just the inequality) is a substantial contribution in its own right. The paper also makes the connection to finite-dimensional Banach spaces explicit through the correspondence with origin-symmetric convex bodies.","major_comments":[],"minor_comments":[{"comment":"The step concluding that each fiber (x+ξ⊥)∩K is a translate of ξ⊥∩K from constant fiber volumes invokes the equality conditions of the Brunn-Minkowski inequality without explicitly stating them; adding a reference or a one-sentence justification would improve the self-containedness of the necessity argument.","section":"Theorem 5.2 (equality case)"},{"comment":"Continuity of Φ_μ on K_n^e under Hausdorff convergence is used implicitly when passing from Blaschke selection to the existence of a minimizer; this follows from uniform convergence of support functions and the fact that the limiting body contains the origin in its interior, but the paper does not state it explicitly.","section":"Theorem 6.3"},{"comment":"The theorem is stated for n≥1, but the proof in the strict case invokes Theorem 6.3, which is stated only for n≥2; the n=1 case is trivial (an even measure on S^0 is the cone-volume measure of a centered interval), but the proof should either split off n=1 or extend Theorem 6.3.","section":"Theorem 7.3"},{"comment":"The implication from unboundedness of Φ_μ(C'_l) to unboundedness of Φ_μ(Q_l) is not written out; it uses h_{C_l}≤h_{Q_l} and the identity Φ_μ(C_l)=Φ_μ(C'_l)+(1/n)log γ, and spelling this out would remove a small gap for the reader.","section":"Theorem 6.3 (contradiction step)"}],"recommendation":"accept","confidential_remarks":"This is the well-known Böröczky–Lutwak–Yang–Zhang solution of the even logarithmic Minkowski problem. The manuscript is careful and the main theorem is landmark. The minor points listed are presentation only and do not affect the correctness of the central claim."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The main theorem is the full characterization: a non-zero even Borel measure on the sphere is a cone-volume measure of an origin-symmetric convex body exactly when it satisfies the subspace concentration condition. That is the real payoff, and the proof delivers it. The variational framework, the symmetrization argument for necessity, and the induction for the equality case all check out. The paper is the published JAMS 2013 article, so the mathematics is already well vetted; I found no circularity and no fitted or invented entities.\n\nWhat is genuinely new: the measure-valued existence theorem, not just the discrete planar case (Stancu) or the necessary inequality for polytopes (He–Leng–Li, Xiong). The subspace concentration condition being both necessary and sufficient is a sharp, checkable characterization. The proof is self-contained modulo standard convex-geometry tools, and the delicate equality case in Theorem 5.2 is handled correctly via Brunn–Minkowski equality conditions. The compactness argument in Theorem 6.3 using John's ellipsoid and the cross-polytope estimate is also sound. I specifically looked for a hidden assumption in the step where equality forces each fiber to be a translate of ξ⊥∩K and the translation locus to be convex; that step is justified.\n\nSoft spots are minor. The paper does not address uniqueness, which is openly stated and is a separate problem. The continuity of the logarithmic functional under Hausdorff convergence is asserted rather than proved explicitly, but this is a standard detail and not a gap. Also, Theorem 7.3 states n≥1 while Theorem 6.3 says n≥2; the n=1 case is trivial, so this is purely cosmetic. The self-citations are background and carry none of the proof.\n\nThe paper is important, careful, and honest about its limitations. The citation pattern is appropriate. This deserves a serious referee—though in practice it already went through one at JAMS and has become a standard reference. I would bring it to a reading group and cite it in my own work. My recommendation: engage with it fully.","headline":"A clean, correct solution of the even logarithmic Minkowski problem; no load-bearing gaps found.","tokens_in":19403,"tokens_out":973,"would_cite":true,"duration_ms":12061,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["52A40"],"pacs":[],"model":"deepseek-v4-flash","headline":"An even measure on the unit sphere is the cone-volume measure of an origin-symmetric convex body exactly when it satisfies the subspace concentration condition, settling the even logarithmic Minkowski problem.","keywords":["cone-volume measure","logarithmic Minkowski problem","Lp-Minkowski problem","subspace concentration condition","origin-symmetric convex body","Brunn-Minkowski inequality","symmetrization","finite-dimensional Banach space"],"falsifier":"The central claim would be falsified by exhibiting a non-zero even Borel measure on $S^{n-1}$ that satisfies the subspace concentration condition yet is not the cone-volume measure of any origin-symmetric convex body; the equality-case decomposition $K=(\\xi^\\perp\\cap K)+C$ is the step most readily probed, for instance by searching for a measure on $S^2$ with equality on a subspace through the origin but no complementary subspace carrying the complementary share, which Theorem 1.1 forbids from being a cone-volume measure.","tokens_in":18473,"feed_emoji":"📐","tokens_out":6972,"duration_ms":64628,"temperature":0.7,"pith_summary":"The paper establishes a complete existence criterion for the even logarithmic Minkowski problem. It shows that a nonzero even Borel measure on the unit sphere is the cone-volume measure of an origin-symmetric convex body in $\\mathbb{R}^n$ precisely when it satisfies the subspace concentration condition: no linear subspace of dimension $k$ may carry more than $k/n$ of the total mass, and equality must be accompanied by a complementary subspace carrying the complementary share. Cone-volume measures are the $p=0$ case of the $L_p$-surface-area measures and are the only ones invariant under all volume-preserving linear maps, so the theorem pins down exactly which sphere data can arise from a finite-dimensional normed space. The paper proves existence by minimizing a logarithmic functional, with the equality case handled through a structural decomposition forced by Brunn\\u2013Minkowski equality. Uniqueness is not treated.","feed_headline":"One sphere condition decides which measures are cone volumes","feed_subtitle":"For symmetric convex bodies, subspace concentration is necessary and sufficient—closing the even logarithmic Minkowski problem.","key_machinery":"The load-bearing objects are the cone-volume measure, defined for a Borel set $\\omega\\subset S^{n-1}$ by $V_K(\\omega)=\\frac1n\\int_{\\nu_K^{-1}(\\omega)} x\\cdot\\nu_K(x)\\,dH^{n-1}(x)$, and the subspace concentration condition stated above. Existence is obtained by minimizing $\\Phi_\\mu(K)=\\int\\log h_K\\,d\\mu$ over origin-symmetric bodies of fixed volume: under strict concentration a minimizer exists via John's ellipsoid and a cross-polytope estimate, and a variational lemma converts the minimizer into a body whose cone-volume measure is $\\mu$. The equality case is the delicate part: symmetrization with respect to a subspace $\\xi$ shows equality in the concentration inequality can happen only if every fiber of $K$ parallel to $\\xi^\\perp$ has the same volume, and the Brunn\\u2013Minkowski equality conditions then force $K$ to be the Minkowski sum $(\\xi^\\perp\\cap K)+C$, so that the measure concentrates on $\\xi$ and a complementary subspace $\\xi'$. This decomposition drives the inductive sufficiency proof.","core_discovery":"The paper's central claim is Theorem 1.1: a non-zero finite even Borel measure $\\mu$ on $S^{n-1}$ is the cone-volume measure of an origin-symmetric convex body in $\\mathbb{R}^n$ if and only if $\\mu$ satisfies the subspace concentration condition. The condition has two parts: for every subspace $\\xi$ with $0<\\dim\\xi