{"id":"8b781ed0-41aa-459b-b1f6-587272b16946","arxiv_id":"2502.05804","paper_version":2,"verdict":"REJECT","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For every nonzero Borel measure not concentrated in a closed hemisphere, there is a convex body whose Lq anisotropic p-torsional measure equals the measure, up to a constant when 0<q<1.","lead":"This paper proves existence of convex bodies whose Lq anisotropic p-torsional measure matches a prescribed Borel measure, for q>1 and 0<q<1. It generalizes known isotropic and q=1 results to anisotropic p-Laplacians.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Lemma 2.4's proof is not a derivation of the claimed variational formula: it differentiates the moving boundary while treating the PDE solution as fixed, so a central identity behind both Theorems 1.1 and 1.3 is left unproved.","rationale":"The reader's weakest_assumption is precisely the proof of Lemma 2.4, and I agree this is the load-bearing point. The proof in §2.2 differentiates the polar upper limit with u fixed and never handles u_t. For elliptic energy functionals the interior variation often cancels in the first variation, so the formula in Lemma 2.4 may itself be correct—indeed q=1 Hadamard formulas have a boundary-only form; the issue is that the paper supplies neither the correct Hadamard derivation nor a reference. Because Lemma 3.1 and Corollary 2.5 are the only bridges from the minimization problems to the measure equation, both Theorems 1.1 and 1.3 are unsupported as written. The factor mismatch between the introductory definition of S_{F,p,q} and (2.15) is a second, smaller obstacle to reading the theorem unambiguously. I would keep the reader's REJECT rather than upgrade to CONDITIONAL, because the written computation appears to differentiate ∫_{Ω_t}F^p(∇u) for the fixed solution u, which is not the same functional as τ_{F,p}([h_t]); without a lemma showing the state derivative cancels or a Hadamard formula, the identity is unproved. The gap is plausibly repairable, and a corrected Hadamard formula should be supplied.","tokens_in":17364,"tokens_out":22667,"duration_ms":244434,"concrete_test":"Re-derive the first variation of τ_{F,p} for Ω_t=[h_t] using the standard material-derivative method: differentiate the anisotropic p-Laplace equation for u_t, integrate by parts, and collect boundary terms. Compare the resulting Hadamard boundary integral with the expression in Lemma 2.4. If the state derivative does not cancel, the fixed-u calculation is wrong; if it cancels, Lemma 2.4 still needs a corrected proof. A useful explicit check is n=2, p=2, F=|·|, q=1, K=B^2, with h_t=1+t cos^2θ; solve -Δu_t=1 in [h_t] to first order in t and compare dτ/dt with ∫_{∂B^2} |∇u|^2 cos^2θ dH = π/4. Any first-order mismatch confirms the gap changes the result.","verdict_should_be":"UNCHANGED","load_bearing_attack":"In §2.2, Lemma 2.4 states lim_{t→0}(τ_{F,p}([h_t])−τ_{F,p}(K))/t = (1/q)∫ f^q dS_{F,p,q}. The proof writes τ_{F,p}([h_t]) = ∫_{S^{n-1}}∫_0^{ρ[h_t](v)} F^p(∇u(ξv))ξ^{n-1}dξdv and differentiates only ρ[h_t] using Lemma 2.2, leaving u fixed. But τ_{F,p}([h_t]) requires the solution u_t of (2.7) in [h_t], so the integrand depends on t; no material derivative of u_t is computed and no Hadamard formula is cited. The rest of the paper invokes Lemma 2.4 at the decisive step: Lemma 3.1 for q>1 and Corollary 2.5/Lemma 4.3 for 0<q<1. Hence both existence theorems rest on an unproved identity. There is also a smaller consistency issue: the definition of S_{F,p,q} in the introduction carries the factor (p−1)/(n(p−1)+p), while (2.15) and the proof of Lemma 2.4 do not; the constant in the claimed identity depends on which definition is meant.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the L_q-Minkowski problem for the anisotropic p-torsional rigidity τ_{F,p}. It claims existence of a convex body K whose L_q anisotropic p-torsional measure S_{F,p,q}(K,·) equals a prescribed nonzero finite Borel measure μ not concentrated in any closed hemisphere, for two regimes: 1<q, q ≠ p/(p-1)+n (Theorem 1.1) and 0<q<1 up to a positive constant (Theorem 1.3). The proof is variational: a first-order expansion of τ_{F,p} under L_q-Minkowski combinations (Lemma 2.4) is used to derive the Euler-Lagrange equation for a minimization problem (q>1) and a polyhedral-approximation argument for 0<q<1. The abstract states that these are existence results beyond the previously known q=1 case.","tokens_in":17604,"tokens_out":5016,"duration_ms":43583,"significance":"If correct, the paper would provide the first existence theorems for the anisotropic p-torsional L_q-Minkowski problem outside q=1, extending results of Li for q=1 and the isotropic p-torsional results of Chen-Zhao-Wang-Zhao and Hu-Liu. The problem is natural and the variational/polyhedral framework is appropriate. The main obstacle is that the central variational formula in Lemma 2.4, on which both existence theorems rest, is not proved; the proof as written omits the dependence of the PDE solution on the varying domain. This is a load-bearing gap, not a presentation issue.","major_comments":[{"comment":"The proof of Lemma 2.4 is not a valid derivation of the claimed variational formula. It writes τ_{F,p}([h_t]) = ∫_{S^{n-1}} ∫_0^{ρ[h_t](v)} F^p(∇u(ξv)) ξ^{n-1} dξ dv, using a function u that appears to be the solution on the fixed body K, and then differentiates only the upper limit ρ[h_t](v) via Lemma 2.2. However τ_{F,p}([h_t]) is defined using the solution u_t of (2.7) on the domain [h_t], and u_t depends on t. No material derivative of u_t is computed, and no Hadamard formula for anisotropic p-torsional rigidity is cited or proved. Consequently the identity lim_{t→0}(τ_{F,p}([h_t])−τ_{F,p}(K))/t = (1/q)∫ f^q dS_{F,p,q} is unproved. This identity is invoked at the decisive points: Lemma 3.1 for q>1 and Corollary 2.5/Lemma 4.3 for the 0<q<1 case. Therefore Theorems 1.1 and 1.3 are not supported by the present proof.","section":"§2.2, Lemma 2.4"},{"comment":"The definition of the L_q anisotropic p-torsional measure is inconsistent across the paper. In the introduction, S_{F,p,q}(K,η) is defined with the prefactor (p−1)/(n(p−1)+p), while equation (2.15) defines it without this prefactor. Lemma 2.4's proof concludes ∫ f^q dS_{F,p,q} using the convention in (2.15). If the introduction's definition is used instead, the variational formula would contain an extra factor. This ambiguity affects the constants in the final existence statements and must be resolved.","section":"§1 and §2.2, definition of S_{F,p,q}"},{"comment":"In the derivation following equation (4.11), the exponent q−1 is replaced by p−1 without justification. Equations (4.12)-(4.14) involve h^{q−1}, but the later identity written as ∫ h^{p−1}(u)f(u)dμ(u) appears with p−1 in several places. The final identification dμ = λ dS_{F,p,q} would be consistent with h^{q−1}, not h^{p−1}. As written, the algebra does not follow; this appears to be a typo, but it occurs in the proof of a main result and must be corrected.","section":"§4, Lemma 4.3"},{"comment":"The statement of Theorem 3.2 assumes that μ is not concentrated on a great subsphere, but the proof uses the stronger assumption that μ is not concentrated in any closed hemisphere. These hypotheses are not equivalent. The uniform-bound argument with the positive constant c0 requires the closed-hemisphere condition. The theorem as stated is stronger than what the proof establishes.","section":"§3, Theorem 3.2"}],"minor_comments":[{"comment":"The hypothesis 'such that ∂K up to set of (n−1)-dimensional Hausdorff measure zero' is grammatically incomplete; the intended regularity condition on ∂K should be stated precisely.","section":"§2.2, Lemma 2.4"},{"comment":"There is a typo in 'Meanehile' in the introduction; it should be 'Meanwhile'.","section":"§1"},{"comment":"The reference to 'Porposition 2.5' in Lemma 2.1 should read 'Proposition 2.5'.","section":"§2.1"},{"comment":"After equation (2.15), the text says 'the Lq anisotropic p-torsional measure, S_{F,p,q}(K,·), of K is a Borel measure', but no proof of Borel measurability is given; this is routine and can be stated as a standard consequence of (2.5).","section":"§2.2, equation (2.15)"}],"recommendation":"reject","confidential_remarks":"The main gap in Lemma 2.4 is substantive: the paper's two existence theorems both depend on a variational formula whose proof omits the domain-dependence of the PDE solution. The discrepancy between the two definitions of S_{F,p,q} and the exponent error in Lemma 4.3 further undermine confidence in the current version. A rejection at this stage seems appropriate; the authors may wish to prove or cite a correct Hadamard formula for anisotropic p-torsional rigidity before resubmitting."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nRead Li-Chen on the L_q-Minkowski problem for anisotropic p-torsional rigidity. The stated goal is clear: establish existence for q>1 and 0<q<1 for the measure S_{F,p,q}, generalizing the isotropic results and the anisotropic q=1 case. The literature review is accurate, and the variational/polyhedral strategy is standard for this area. The statements are plausible, and if the main variational formula were solid, the rest of the arguments would likely go through.\n\nThe problem: Lemma 2.4 is load-bearing and its proof is not valid. They define τ_{F,p}([h_t]) as the integral of F^p(∇u) over [h_t], but u in that integral must be the solution of (2.7) in [h_t], which depends on t. The proof differentiates only the radial limit ρ[h_t](v), treating u as fixed. No material derivative of u_t is computed, and no Hadamard-style formula for τ_{F,p} under L_q-Minkowski deformations is cited. The limit they get is the derivative of the domain term only. That gives the right hand side formally, but there is no justification that the interior variation vanishes. Both Theorem 1.1 and Theorem 1.3 invoke Lemma 2.4 at the decisive step (via Lemma 3.1 and Corollary 2.5/Lemma 4.3). So the central existence claims rest on an unproved identity.\n\nThere is also a smaller consistency issue: the measure S_{F,p,q} is defined with the factor (p−1)/(n(p−1)+p) in the introduction, but (2.15) and the proof of Lemma 2.4 use the measure without that factor. That changes the constant in the variational formula; presumably fixable, but it needs cleaning up.\n\nThat said, I don't see circularity or fitted parameters. The reliance on the author's earlier q=1 result [29] is legitimate, and the continuity/homogeneity facts from [9] are appropriate. The gap looks repairable: a correct Hadamard formula for anisotropic p-torsional rigidity might already be derivable from the known shape derivatives for the p-torsion, or from the author's q=1 work with a bit more care. But as submitted, the main theorems are unsupported.\n\nFor peer review: I'd send it to a referee. The problem is natural, the literature is handled responsibly, and the gap is specific enough that a competent referee could tell whether it's fixable. It deserves the referee time, just not acceptance in this form.\n\nWould I cite it? Not until the variational formula is fixed. But I'd bring it to a reading group as a case study in how the moving domain and the solution variation have to be handled together.","headline":"New existence results for a natural L_q-Minkowski problem, but the core variational formula is unproved as written: the proof fixes the PDE solution while the domain moves.","tokens_in":18182,"tokens_out":3523,"would_cite":false,"duration_ms":30395,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35N25","52A20","53C21","31A15"],"pacs":[],"model":"deepseek-v4-flash","headline":"For every admissible Borel measure there is a convex body whose Lq anisotropic p-torsional measure equals it, for q>1 and 0<q<1.","keywords":["Lq-Minkowski problem","anisotropic p-torsional rigidity","anisotropic p-Laplacian","convex bodies","Wulff shape","Borel measure","variational method"],"falsifier":"Take $K$ the unit ball, $F(\\xi)=|\\xi|$, $p=2$, and some $q\\neq1$ with a nonconstant $f$, and numerically solve the Poisson problem in the deformed body $[h_t]$ for small $t$ to evaluate $\\tau_{F,p}([h_t])$ directly; if the difference quotient does not converge to $\\frac1q\\int f^q\\,dS_{F,p,q}(K,\\cdot)$, then Lemma 2.4 is false and the main theorems no longer have a proved basis.","tokens_in":17112,"feed_emoji":"🔷","tokens_out":7689,"duration_ms":68232,"temperature":0.7,"pith_summary":"This paper proves existence for the $L_q$-Minkowski problem of anisotropic $p$-torsional rigidity: given any nonzero finite Borel measure on the unit sphere that is not concentrated in a closed hemisphere, there is a convex body containing the origin whose $L_q$ anisotropic $p$-torsional measure is exactly that measure, for $1<q\\neq \\frac{p}{p-1}+n$, and for $0<q<1$ the same statement holds up to a positive multiplicative constant. The $L_q$ anisotropic $p$-torsional measure is a geometric measure built from the solution of an anisotropic $p$-Laplace boundary-value problem, so the result says that these PDE-generated measures are rich enough to realize every admissible sphere measure. A sympathetic reader would care because it extends the classical Minkowski problem—determining a convex body from a surface measure—to a family of measures that arise naturally from torsion of anisotropic materials, and because the $q=1$ case is the only one previously understood. If correct, these are the first existence results outside the $q=1$ case for this class of problems.","feed_headline":"Existence proved for anisotropic p-torsional Minkowski problem","feed_subtitle":"For q>1 and 0<q<1, every admissible measure is the Lq anisotropic p-torsional measure of some convex body.","key_machinery":"The central object is the $L_q$ anisotropic $p$-torsional measure $S_{F,p,q}(K,\\eta)$, defined by $S_{F,p,q}(K,\\eta)=\\int_{g_K^{-1}(\\eta)}\\langle x,g_K(x)\\rangle^{1-q}F^p(\\nabla u(x))\\,d\\mathcal H^{n-1}(x)$, where $u$ solves the anisotropic $p$-Laplace problem in $K$ and $g_K$ is the Gauss map. The identity carrying the argument is the variational formula of Lemma 2.4: for $h_t=(h_K^q+t f^q)^{1/q}$, the one-sided derivative of $\\tau_{F,p}([h_t])$ at $t=0$ equals $\\frac1q\\int f^q\\,dS_{F,p,q}(K,\\cdot)$. This formula converts the geometric minimization into the measure equation; for $0<q<1$ the key machinery is instead the strictly concave functional $\\Phi_{f,\\mu}(\\xi)=\\int (f(u)-\\xi\\cdot u)^q\\,d\\mu(u)$ and its unique maximizer over the Aleksandrov body, which yields the measure equation by an implicit-function argument.","core_discovery":"The paper's central claim is that the solution set of the anisotropic $p$-torsional Minkowski problem is essentially as large as in the classical case. Specifically, Theorem 1.1 asserts that for $1<p<\\infty$ and $1<q\\neq \\frac{p}{p-1}+n$, every nonzero finite Borel measure $\\mu$ on $S^{n-1}$ not concentrated in a closed hemisphere is of the form $S_{F,p,q}(K,\\cdot)$ for some convex body $K\\in\\mathcal K_o^n$. Theorem 1.3 asserts the same for $0<q<1$ in the proportional form $d\\mu=\\lambda\\,dS_{F,p,q}(K,\\cdot)$ with a positive constant $\\lambda$. The proof constructs $K$ as a minimizer of a log-type functional involving $\\tau_{F,p}$, then uses the variational formula for the derivative of $\\tau_{F,p}$ along $L_q$-Minkowski combinations; for small $q$ it instead uses a concave maximization trick on polyhedral (discrete-measure) approximations and passes to the limit.","pith_inferences":["If the variational formula is correct, the same minimization scheme should extend to $q<0$ whenever the functional $\\Psi_{F,p,q}$ remains coercive; the paper does not treat that range.","The formal structure of the measure density—$h^{1-q}$ times a PDE-generated density—mirrors the electrostatic $q$-capacitary Minkowski problem, so techniques from either setting may transfer to the other.","A direct numerical check of Lemma 2.4 on radial bodies would settle whether the omitted interior variation of the PDE solution is benign; until then, the $q\\neq1$ existence results rest on that unverified formula."],"forward_implications":["The $q=1$ case—the ordinary anisotropic $p$-torsional Minkowski problem—sits inside Theorems 1.1 and 1.3 as a limiting special case, so the result unifies and extends the known existence theory.","For $F(\\xi)=\\sum_k |\\xi_k|$ and $p=2$, the theorems reproduce the known $L_p$ torsional-rigidity Minkowski problem results, giving a single framework that contains them.","The minimizer produced by the variational proof is automatically a convex body whose $L_q$ anisotropic $p$-torsional measure is the prescribed measure, so existence comes with a constructive variational characterization.","The $0<q<1$ result holds for general Borel measures, not just discrete ones, because the discrete case is dense in the weak topology under the hypothesis that the measure is not concentrated in a closed hemisphere."],"supporting_citations":[{"why":"Introduces the anisotropic p-torsional measure, the q=1 problem, and the variational/continuity facts the paper extends.","marker":"[29]"},{"why":"Supplies the sharp upper bound on $\\tau_{F,p}$ (Lemma 2.3) used to keep minimizing sequences from degenerating.","marker":"[9]"},{"why":"Provides the $L_q$-Minkowski combination and the variational framework that the paper adapts to torsional rigidity.","marker":"[35]"},{"why":"Supplies convex-geometric background, the Aleksandrov/Wulff body construction, Blaschke selection, and the denseness of discrete measures used in the approximation argument.","marker":"[38]"},{"why":"Contains Lemma 4.1, the strict concavity and unique sup point for the $0<q<1$ functional.","marker":"[25]"},{"why":"Supplies the approximation scheme that passes from discrete to general Borel measures in the $0<q<1$ case.","marker":"[11]"},{"why":"Contains the anisotropic Pohozaev identity used to derive the integral representation of $\\tau_{F,p}$.","marker":"[2]"},{"why":"Is the p-torsional rigidity special case whose $0<p<1$ theorem is recovered by Theorem 1.3.","marker":"[5]"},{"why":"Is the classical torsion special case recovered when $p=2$ and $q=1$.","marker":"[6]"}],"fun_headline_variants":["Existence for anisotropic p-torsional Minkowski in both q ranges","Anisotropic p-torsional Minkowski: existence for all admissible measures","For q<1 and q>1, anisotropic p-torsional Minkowski has solutions","Minkowski problem with anisotropic p-torsion: existence proved"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The argument stands on the variational formula in Lemma 2.4, which computes the derivative of the torsional rigidity by changing only the domain and holding the PDE solution fixed; if the solution's variation inside the domain contributes to this derivative, the formula fails and the existence theorems lose their support.","fun_headline_variants_meta":{"raw":{"variants":["Existence for anisotropic p-torsional Minkowski in both q ranges","Anisotropic p-torsional Minkowski: existence for all admissible measures","For q<1 and q>1, anisotropic p-torsional Minkowski has solutions","Minkowski problem with anisotropic p-torsion: existence proved"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000997,"raw_usage":{"total_tokens":4164,"prompt_tokens":832,"completion_tokens":3332,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":448,"completion_tokens_details":{"reasoning_tokens":3246}},"tokens_in":448,"tokens_out":3332,"duration_ms":25738,"temperature":1.0,"reasoning_tokens":3246,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-08T17:52:47.298734+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take $K$ the unit ball, $F(\\xi)=|\\xi|$, $p=2$, and some $q\\neq1$ with a nonconstant $f$, and numerically solve the Poisson problem in the deformed body $[h_t]$ for small $t$ to evaluate $\\tau_{F,p}([h_t])$ directly; if the difference quotient does not converge to $\\frac1q\\int f^q\\,dS_{F,p,q}(K,\\cdot)$, then Lemma 2.4 is false and the main theorems no longer have a proved basis.","supporting_citations":[{"cited_title":"Minkowski problem of anisotropic p-torsional rigidity","cited_arxiv_id":"2501.00687","evidence_quote":"Introduces the anisotropic p-torsional measure, the q=1 problem, and the variational/continuity facts the paper extends."},{"cited_title":"Della Pietra and N","cited_arxiv_id":null,"evidence_quote":"Supplies the sharp upper bound on $\\tau_{F,p}$ (Lemma 2.3) used to keep minimizing sequences from degenerating."},{"cited_title":"Lutwak, The Brunn-Minkowski-Firey theory I: Mixed volumes and the M inkowski problem , J","cited_arxiv_id":null,"evidence_quote":"Provides the $L_q$-Minkowski combination and the variational framework that the paper adapts to torsional rigidity."},{"cited_title":"Schneider, Convex Bodies: The Brunn-Minkowski theory , 2nd edn, Cambridge Univ","cited_arxiv_id":null,"evidence_quote":"Supplies convex-geometric background, the Aleksandrov/Wulff body construction, Blaschke selection, and the denseness of discrete measures used in the approximation argument."},{"cited_title":"Jian and J","cited_arxiv_id":null,"evidence_quote":"Contains Lemma 4.1, the strict concavity and unique sup point for the $0<q<1$ functional."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the approximation scheme that passes from discrete to general Borel measures in the $0<q<1$ case."},{"cited_title":"Bianchin and G","cited_arxiv_id":null,"evidence_quote":"Contains the anisotropic Pohozaev identity used to derive the integral representation of $\\tau_{F,p}$."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Is the p-torsional rigidity special case whose $0<p<1$ theorem is recovered by Theorem 1.3."},{"cited_title":"Chen and Q","cited_arxiv_id":null,"evidence_quote":"Is the classical torsion special case recovered when $p=2$ and $q=1$."}],"review_version":1}