{"id":"bc09fde3-7cd9-49c5-aaa6-fb7d9c76f259","arxiv_id":"2412.13651","paper_version":1,"verdict":"REJECT","confidence":"MODERATE","novelty_score":4.0,"correctness_risk":"high","formal_verification":"none","parameter_count":0,"one_line_summary":"For p,q>1, an origin-symmetric convex body is claimed to exist whose normalized Lp Gaussian dual curvature measure matches any given even density measure, up to total mass scaling.","lead":"This paper proves a new existence result for the even Lp Gaussian dual Minkowski problem, a geometric question about matching a prescribed measure on the sphere by a curvature-type measure of a symmetric convex body. The proof extends the variational method of Feng, Hu and Xu from the Gaussian dual case to the Lp setting with p>1.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 1.1 assumes only an even density, but its only proof (Theorem 4.2) requires a two-sided bound; the missing lower bound is used essentially in the compactness argument, so the main theorem is not derived as stated.","rationale":"The reader's weakest_assumption identifies the same load-bearing gap: Theorem 1.1 is stated for arbitrary even densities, while Theorem 4.2 only proves existence under a two-sided bound. I checked the proof of Theorem 4.2 and confirmed that the lower bound on the density is used to obtain a uniform upper bound on the maximizing sequence; without it, the argument collapses. The Lagrange multiplier step in Theorem 4.1 is informal but repairable through the standard necessary condition for constrained extrema, since the constraint derivative is nonzero for positive perturbations; the density bounds, by contrast, are an unproved hypothesis essential to the existence proof. The exact-vs-normalized mismatch is a statement issue but not fatal because Theorem 1.1 explicitly states the normalized form. Therefore the single most load-bearing concern is the missing boundedness assumption, and it lands. The verdict REJECT remains appropriate: as written, the main theorem is not established. A revised version that adds the two-sided density bound (or supplies a nontrivial approximation argument) could change the verdict.","tokens_in":12537,"tokens_out":6005,"duration_ms":58743,"concrete_test":"Take n=2, p=q=2, and let dμ(u)=|u_1|^2 du on S^1, an even density with infimum 0. For the sequence of origin-symmetric convex bodies K_i obtained by rounding the rectangle [-R_i,R_i]×[-1,1] with R_i→∞, compute Φ(K_i)= -∫ h_{K_i}^2 |u_1|^2 du /(p|μ|) and the Gaussian dual quermassintegral ~V_{γ,2}(K_i). Direct calculation will show Φ(K_i) remains bounded below while R_i→∞, so the key estimate in Theorem 4.2 fails for every finite M0. This confirms that the missing lower bound is not a harmless omission but a hypothesis on which the stated compactness argument depends.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim is Theorem 1.1: for p,q>1, any nonzero finite Borel measure with an even density has a normalized solution of the Lp Gaussian dual Minkowski problem. The only existence proof, Theorem 4.2, assumes the density satisfies 1/M0 ≤ g(u) ≤ M0. That assumption is absent from Theorem 1.1, and the proof of Theorem 4.2 uses it essentially. To show the maximizing sequence K_i is bounded, the authors let 2R_i be the length of the longest segment in K_i and use R_i|⟨v_i,u⟩| ≤ h_{K_i}(u) to estimate Φ(K_i) ≤ - R_i^p/(p|μ|) ∫ |⟨v_i,u⟩|^p g(u) du. The lower bound g ≥ 1/M0 then gives Φ(K_i) ≤ - R_i^p/(p M0 |μ|) ∫ |⟨v_i,u⟩|^p du, which tends to -∞ if R_i→∞. If g is an even density without a positive lower bound, the integral ∫ |⟨v_i,u⟩|^p g(u) du can be arbitrarily small relative to |μ|, so the contradiction fails and boundedness of the maximizing sequence is not established. The cited lower bound for h_{K_i}, imported from [18], likewise depends on the measure's lower bound and is not proved here. Thus Theorem 1.1 does not follow from the supplied arguments. The inconsistency of Theorem 4.3, which restates the result for p,q>0, reinforces that the hypotheses were not carefully aligned.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the even L_p Gaussian dual Minkowski problem. It introduces an L_p Gaussian dual curvature measure, proves a first-variation formula for the Gaussian dual quermassintegral, and then uses a variational method to claim existence of origin-symmetric solutions for p,q>1. The main theorem (Theorem 1.1) states that for any nonzero finite Borel measure with an even density, the normalized prescribed measure is realized as the normalized L_p Gaussian dual curvature measure of some origin-symmetric convex body.","tokens_in":12834,"tokens_out":7577,"duration_ms":69985,"significance":"If the main theorem were correctly established, the paper would provide a meaningful extension of the Gaussian dual Minkowski problem to the L_p setting, with a new curvature measure and a variational proof. The derivation of the variational formula (3.5) and the optimization framework are potentially useful. However, as written, the proof does not support the advertised theorem: the only existence argument assumes stronger hypotheses than Theorem 1.1 states, and the Lagrange multiplier step is not rigorously justified. The central claim therefore remains unproved, and the paper needs substantial revision before it can be considered for publication.","major_comments":[{"comment":"Theorem 4.2 assumes the density satisfies 1/M0 ≤ g(u) ≤ M0, while Theorem 1.1 assumes only an even density f with no lower or upper bound. The boundedness argument for the maximizing sequence uses the lower bound essentially: the inequality Φ(K_i) ≤ -R_i^p/(p M0 |μ|) ∫ |⟨v_i,u⟩|^p du (page 13) is obtained from g ≥ 1/M0. If g is an arbitrary even density, the integral ∫ |⟨v_i,u⟩|^p g(u) du can be arbitrarily small, so the contradiction with (4.7) fails. No approximation of a general even density by densities satisfying two-sided bounds is provided. Thus Theorem 1.1 does not follow from Theorem 4.2.","section":"§4.2, Theorem 4.2 vs. Theorem 1.1"},{"comment":"The proof asserts that a uniform positive lower bound, min_v h_{K_i}(v) ≥ M1, 'has been illustrated in [18]'. This is a load-bearing step in the compactness argument, but the precise statement and hypotheses of the cited result are not given, and it is not verified that those hypotheses hold in the setting of Theorem 4.2. If the cited bound itself depends on a two-sided density bound, then this is another unstated assumption that must be made explicit and checked against Theorem 1.1.","section":"§4.2, lower bound for h_{K_i}"},{"comment":"The proof differentiates Γ(t,λ) = Φ(h_t) + λ(~V_{γ_{n,q}}([h_t]) - |μ|) at t=0 and sets this derivative to zero. However, the path h_t = (h_{K_0}^p + t g^p)^{1/p} does not generally satisfy the constraint ~V_{γ_{n,q}}([h_t]) = |μ| for t≠0, so the derivative of the objective along this infeasible path at a constrained maximum need not vanish. A rigorous derivation requires either an application of the Lagrange multiplier rule (with a regularity check on the constraint functional) or a reparametrization of the path to stay on the constraint manifold. Without such an argument, equation (4.5) is not established.","section":"§4.1, Theorem 4.1, equation (4.5)"},{"comment":"Theorem 4.3 states the result for 'p, q > 0', whereas the whole proof, including the definition of the functional Φ in (4.1) and the variational formula, requires p > 1. This inconsistency indicates that the hypotheses across the theorems are not carefully aligned and leaves the claimed generalization unsupported.","section":"§4.3, Theorem 4.3"}],"minor_comments":[{"comment":"The abstract contains grammatical errors and an incomplete sentence ('The even Gaussian dual Minkowski problem studied by Feng, Hu and Xu, In this paper...'). It should be rewritten.","section":"Abstract"},{"comment":"The text states that it is essential to demonstrate weak convergence of ~C_{p,γ_{n,q}}(K,·) with respect to the Hausdorff metric and absolute continuity with respect to the surface area measure, but no proof of these statements is given in the paper. Either the proof should be supplied or the assertion should be removed.","section":"§3, end of section"},{"comment":"The symbol g is used both for the density of μ (in Theorem 4.2) and as an arbitrary test function in C_e^+ (in Theorem 4.1). This overloading is confusing and should be resolved by using different letters.","section":"§4.1, notation"},{"comment":"Lemma 3.1 is imported verbatim from the authors' earlier paper [40]. While citing an external lemma is acceptable, the dependency should be stated clearly in the introduction, and the lemma should be quoted with its full hypotheses, since the later proofs rely critically on it.","section":"§3, Lemma 3.1"}],"recommendation":"major_revision","confidential_remarks":"The paper addresses a timely and interesting problem, and the variational formula for the new L_p Gaussian dual curvature measure is a useful contribution. However, the main existence theorem is not proved as stated: the compactness argument requires a two-sided density bound, and the Lagrange multiplier step in Theorem 4.1 is not rigorous. These are not cosmetic issues; they affect the core claim. I recommend major revision rather than outright rejection because the approach is promising and the gaps, while substantial, may be addressable with a careful rewriting of the proof or a more honest statement of the theorem. The authors should either prove Theorem 1.1 under the stated hypotheses (e.g., via an approximation argument) or modify the theorem to include the hypotheses actually used."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Quick take: this is an honest extension of the even Gaussian dual Minkowski problem to the Lp setting, with a new measure and a standard variational existence proof. But the main theorem as printed is not supported. The only existence proof, Theorem 4.2, assumes 1/M0 ≤ g ≤ M0; Theorem 1.1 only assumes an even density. The lower bound is used essentially in the boundedness argument for the maximizing sequence: without it the integral ∫ |⟨v_i,u⟩|^p g(u) du can be arbitrarily small, and the contradiction Φ(K_i) → −∞ fails. The cited lower-bound argument from [18] also depends on such a bound. So the central theorem is not derived.\n\nSecond, Theorem 4.1’s Lagrange multiplier step differentiates ht = (h_{K0}^p + t g^p)^{1/p} while the constraint is Ṽ_{γn,q}([ht]) = |μ|, but ht does not generally preserve the volume. Setting ∂Γ/∂t = 0 at t=0 is a standard heuristic, not a valid Lagrange multiplier argument. It might be salvageable by reparametrizing the perturbation, but no justification is given.\n\nAlso, the introduction asks for exact equality μ = C̃_{p,γn,q}, while Theorem 1.1 gives normalized equality μ/|μ| = C̃/|C̃|. That mismatch matters. Theorem 4.3 then restates the result for p,q>0, contradicting p,q>1 elsewhere. These are real hypothesis-alignment problems.\n\nCredit where due: the Lp Gaussian dual curvature measure is new, the variational formula (Theorem 3.1) looks correct with the standard dominated-convergence argument, and the paper is honest about relying on [18] and [40] for the heavy lifting. The self-citations are fine—those are published external lemmas. The English is rough, but that is not the blocker.\n\nFor convex geometers working on Gaussian/Lp Minkowski problems, this is a plausible and field-relevant extension. With the density bounds added to Theorem 1.1, or the statement restricted to 1/M0 ≤ f ≤ M0, and a legitimate Lagrange multiplier step, the result would likely hold. As is, a serious referee should require major revision or reject the preprint. It still deserves referee time because the core idea is sound and repairable.","headline":"A repairable parameter-extension paper whose main theorem is not proved as stated: Theorem 1.1 omits the two-sided density bounds that the only proof, Theorem 4.2, uses essentially.","tokens_in":13390,"tokens_out":1959,"would_cite":false,"duration_ms":19566,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["52A40","52A38","35J96"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves that for $p,q>1$, any nonzero finite Borel measure on the sphere with an even density is, up to normalization, the $L_p$ Gaussian dual curvature measure of an origin-symmetric convex body.","keywords":["Lp Gaussian dual curvature measure","Lp Gaussian dual Minkowski problem","Monge-Ampere equation","convex body","even density","variational method","Wulff shape","Gaussian dual quermassintegral"],"falsifier":"Take the even density $f(u)=|u_1|$ on the unit circle in the plane with $p=q=2$ and check whether the constrained maximization of $\\Phi$ is attained by some origin-symmetric convex body. If no such body exists, the theorem as stated is false; if one does, this example would show the boundedness hypothesis is not always necessary.","tokens_in":12263,"feed_emoji":"📐","tokens_out":14669,"duration_ms":119238,"temperature":0.7,"pith_summary":"This paper proves an existence theorem in convex geometry: for any two exponents $p,q>1$, every nonzero finite Borel measure on the unit sphere that has an even density function can be matched, up to normalization, by the $L_p$ Gaussian dual curvature measure of an origin-symmetric convex body. The statement matters because it solves the even $L_p$ Gaussian dual Minkowski problem, a Monge-Ampere-type equation in Gaussian probability space that mixes the $L_p$ Brunn-Minkowski theory with dual curvature measures. The proof is variational: it maximizes a natural functional over origin-symmetric convex bodies with a fixed Gaussian dual quermassintegral, then reads the optimality condition as the desired measure equation. If the theorem is correct, it extends the even Gaussian dual Minkowski problem from the $p=1$ case to all $p>1$ and gives solvability of the associated spherical Monge-Ampere equation for even data.","feed_headline":"For any p and q above 1, every even sphere density is realizable","feed_subtitle":"For p,q>1, an origin-symmetric convex body realizes any even density, up to normalization.","key_machinery":"The load-bearing object is the $L_p$ Gaussian dual curvature measure, defined for a convex body $K$ by $\\tilde C_{p,\\gamma_{n,q}}(K,\\eta)=\\int_{\\nu_K^{-1}(\\eta)} \\langle x,\\nu_K(x)\\rangle^{1-p}|x|^{q-n}e^{-|x|^2/2}\\,dH^{n-1}(x)$. The mechanism that carries the argument is the variational identity (3.5): if $h_t=(h_K^p+t f^p)^{1/p}$ is a Wulff-shape deformation, then $\\frac{d}{dt}\\tilde V_{\\gamma_{n,q}}([h_t])|_{t=0}=\\frac{1}{p}\\int f(v)^p\\,d\\tilde C_{p,\\gamma_{n,q}}(K,v)$. This identity turns the geometric existence problem into an optimization problem, because the desired normalized measure equation appears as the Euler-Lagrange equation for maximizing $\\Phi(K)$ under a fixed Gaussian dual quermassintegral. Compactness is supplied by a standard selection theorem for convex bodies once the maximizing sequence is shown to be uniformly bounded below and above; the proof of that boundedness is where the density's positive lower and upper bounds enter.","core_discovery":"The central claim is Theorem 1.1: for $p,q>1$, if $\\mu$ is a nonzero finite Borel measure on $S^{n-1}$ with even density $f$, then there exists an origin-symmetric convex body $K$ such that $\\mu/|\\mu| = \\tilde C_{p,\\gamma_{n,q}}(K,\\cdot)/|\\tilde C_{p,\\gamma_{n,q}}(K,\\cdot)|$. The measure $\\tilde C_{p,\\gamma_{n,q}}(K,\\cdot)$ is built from the Gaussian dual quermassintegral $\\tilde V_{\\gamma_{n,q}}(K)=\\int_K e^{-|x|^2/2}|x|^{q-n}\\,dH^{n-1}(x)$ through an $L_p$ variational formula: under the Wulff-shape deformation $h_t=(h_K^p+t f^p)^{1/p}$, the derivative of $\\tilde V_{\\gamma_{n,q}}$ at $t=0$ equals $(1/p)\\int f^p\\,d\\tilde C_{p,\\gamma_{n,q}}(K,\\cdot)$. The proof introduces the functional $\\Phi(K)=-(p|\\mu|)^{-1}\\int h_K^p\\,d\\mu$ and maximizes it over origin-symmetric convex bodies subject to $\\tilde V_{\\gamma_{n,q}}(K)=|\\mu|$; an optimizing sequence is shown to converge to a body $K_0$, and the first-order condition yields the normalized equality. The paper also records the smooth-case equivalence with a Monge-Ampere equation on $S^{n-1}$.","pith_inferences":["The proof's boundedness step suggests that the theorem is proved at the generality of densities bounded above and below by positive constants; whether the statement with arbitrary even densities holds is a natural testable question.","Because the conclusion is normalization-invariant, one can also read the result as a statement about probability measures: every even probability density is the normalized $L_p$ Gaussian dual curvature measure of some symmetric body.","Extrapolating from the variational structure, the most delicate regime is likely $p$ near $0$ or $1$, where the $L_p$ deformation loses coercivity and the logarithmic ($p=0$) counterpart would require different estimates."],"forward_implications":["For any even density and any $p,q>1$, the normalized $L_p$ Gaussian dual Minkowski problem has an origin-symmetric solution.","In the smooth case the existence result is equivalent to solvability of a Monge-Ampere equation on the sphere with an even right-hand side.","The variational method provides a template: the same functional and compactness argument can be adapted to related Gaussian dual Minkowski-type problems.","The realized curvature measure is matched only up to a normalizing constant, so the theorem does not prescribe the total mass of $\\tilde C_{p,\\gamma_{n,q}}(K,\\cdot)$."],"supporting_citations":[{"why":"Supplies the Gaussian dual quermassintegral, its convergence lemma, and the earlier even Gaussian dual Minkowski problem that this paper extends.","marker":"[18]"},{"why":"Introduces the Wulff-shape/logarithmic-family deformation technique and the dual curvature measures that the $L_p$ Gaussian dual measure generalizes.","marker":"[31]"},{"why":"Provides Lemma 3.1, the $L_p$ variational formula for radial functions of Wulff shapes, used in computing the derivative of the Gaussian dual quermassintegral.","marker":"[40]"},{"why":"Supplies the compactness and convergence theorems for convex bodies used to extract a maximizing sequence limit.","marker":"[52]"}],"fun_headline_variants":["For p,q>1, every even density on the sphere is realizable","Even L_p Gaussian dual Minkowski: existence for p,q>1","Any even measure on S^n realized by a convex body when p,q>1","O-symmetric solution for even L_p Gaussian dual Minkowski"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof's compactness step requires the even density to be bounded above and below by positive constants, an assumption the main theorem's statement does not include.","fun_headline_variants_meta":{"raw":{"variants":["For p,q>1, every even density on the sphere is realizable","Even L_p Gaussian dual Minkowski: existence for p,q>1","Any even measure on S^n realized by a convex body when p,q>1","O-symmetric solution for even L_p Gaussian dual Minkowski"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000547,"raw_usage":{"total_tokens":2598,"prompt_tokens":910,"completion_tokens":1688,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":526,"completion_tokens_details":{"reasoning_tokens":1606}},"tokens_in":526,"tokens_out":1688,"duration_ms":12861,"temperature":1.0,"reasoning_tokens":1606,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T12:56:11.808078+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take the even density $f(u)=|u_1|$ on the unit circle in the plane with $p=q=2$ and check whether the constrained maximization of $\\Phi$ is attained by some origin-symmetric convex body. If no such body exists, the theorem as stated is false; if one does, this example would show the boundedness hypothesis is not always necessary.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the Gaussian dual quermassintegral, its convergence lemma, and the earlier even Gaussian dual Minkowski problem that this paper extends."},{"cited_title":"Huang, E","cited_arxiv_id":null,"evidence_quote":"Introduces the Wulff-shape/logarithmic-family deformation technique and the dual curvature measures that the $L_p$ Gaussian dual measure generalizes."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides Lemma 3.1, the $L_p$ variational formula for radial functions of Wulff shapes, used in computing the derivative of the Gaussian dual quermassintegral."},{"cited_title":"Schneider","cited_arxiv_id":null,"evidence_quote":"Supplies the compactness and convergence theorems for convex bodies used to extract a maximizing sequence limit."}],"review_version":1}