{"id":"6dc0716f-34c0-40d9-a7af-b7ba31b98776","arxiv_id":"2501.00687","paper_version":3,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For the Finsler p-Laplacian torsion problem, a nonzero finite measure is an anisotropic p-torsional measure of a convex body iff its centroid is the origin and it is not concentrated on a closed hemisphere; the log version holds under a subspace mass inequality.","lead":"For the Finsler p-Laplace equation on a convex shape, this paper asks which measures on the unit sphere can be realized as the 'p-torsional' boundary measure of the shape, and gives complete conditions in the standard case plus a sufficient condition in the logarithmic case. A reader interested in convex geometry or elliptic PDE will see an anisotropic analogue of classical Minkowski-type problems with a variational proof.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Lemma 3.13's proof differentiates the moving domain while freezing the PDE solution; the omitted t-derivative of u_t is load-bearing for all variational theorems.","rationale":"The reader's weakest assumption identifies exactly the load-bearing gap: Lemma 3.13 is the only bridge between the torsional rigidity functional and the measure S_{F,p}, and the proof as written computes the derivative of a domain integral with the solution frozen, rather than the derivative of the actual PDE-constrained functional. I read the rest of the paper as depending on this lemma in every variational step, so the concern is not peripheral. I do not conclude that the theorem is false: the shape-derivative formula is likely correct, and the paper has independent structural support from the classical torsional case, but the submitted proof is incomplete at a load-bearing point. Therefore the appropriate verdict is unchanged from the reader's CONDITIONAL; no grounds for acceptance or rejection are established by the submitted text.","tokens_in":29551,"tokens_out":10316,"duration_ms":108720,"concrete_test":"Recompute Lemma 3.13 with u_t = u_{[h_t]} treated as t-dependent: differentiate τ_{F,p}([h_t]) through a domain-velocity field with normal component f and explicitly retain the term involving ∂_t u_t; then use the linearized PDE for u_t and the boundary condition to verify cancellation. If cancellation holds, the formula in Lemma 3.13 stands but must be rewritten with a full shape-derivative proof and stated regularity hypotheses for passing from C^{2,α} bodies to convex limit bodies. If cancellation fails, the missing term changes the coefficient or adds a boundary contribution, and the variational proofs of Theorems 1.1-1.3 collapse.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Lemma 3.13 claims lim_{t→0}(τ_{F,p}([h_t])-τ_{F,p}(K))/t = ∫_{S^{n-1}} f dS_{F,p}(K). Its proof writes τ_{F,p}([h_t]) = ∫_{S^{n-1}} ∫_0^{ρ[h_t](v)} F^p(∇u(ξv)) ξ^{n-1} dξ dv and differentiates only the upper limit, using the solution u on K. But for the Wulff shape [h_t] the relevant solution is u_t, not u, so differentiating G_t(v) also produces the interior term ∫_0^{ρ_K(v)} ∂_t[F^p(∇u_t(ξv))] ξ^{n-1} dξ. The proof neither computes nor bounds this term; its dominated-convergence estimate uses the frozen F^p(∇u(r_K(v))). A standard shape-derivative computation with normal velocity f shows the interior term can cancel through the PDE for u_t and the boundary condition u_t=0, so the formula is plausible, but the paper does not supply that argument. Lemma 3.13 is invoked in Lemma 3.14, Lemma 4.3, Lemma 5.4, and Lemma 5.5, so Theorems 1.1, 1.2, and 1.3 all depend on this missing step. A related gap is the use of the C^{2,α} gradient bounds from Lemma 3.8 for bodies obtained by Blaschke selection, which are only known to be convex.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the Minkowski problem for the anisotropic p-torsional measure S_{F,p}(K, ·) and the cone anisotropic p-torsional measure τ^log_{F,p}(K, ·), both generated by the solution of the Finsler p-Laplacian Dirichlet problem. The main results are: Theorem 1.1, a full characterization of measures that arise as S_{F,p}(K, ·) for some convex body, namely non-vanishing, not concentrated on a closed hemisphere, and having barycenter at the origin; Theorem 1.2, existence for the log-Minkowski problem for discrete measures with supports in general position; and Theorem 1.3, existence for general measures satisfying a subspace mass inequality (for 2 ≤ p < ∞). The proofs follow a variational strategy: a first-variation formula for the anisotropic p-torsional rigidity, a minimization argument over convex bodies, and an approximation argument for the log-Minkowski case. The paper also establishes basic properties such as homogeneity, translation invariance, monotonicity, and weak continuity of the new measures.","tokens_in":29919,"tokens_out":6315,"duration_ms":63672,"significance":"If the proofs are correct, Theorem 1.1 provides a complete and clean analogue of the classical Minkowski existence theorem for a new anisotropic torsional measure, and Theorems 1.2–1.3 extend log-Minkowski existence to a non-symmetric setting. The problem formulation is natural and the variational framework is well chosen, building on the established lines of Jerison, Colesanti–Fimiani, Zhu, and Hu. The paper is also honest about the hypotheses: the subspace mass inequality in Theorem 1.3 is an explicit assumption, and there is no circularity in the existence arguments. The main obstacles are technical: the first-variation lemma is under-proved, and several limiting steps are applied to bodies with less regularity than the invoked lemmas assume. These gaps are localized and appear fixable, so the underlying contribution is potentially significant.","major_comments":[{"comment":"Lemma 3.13 computes the derivative of τ_F,p([h_t]) by differentiating only the upper limit of the polar-coordinate integral ∫_0^{ρ[h_t](v)} F^p(∇u(ξ v)) ξ^{n-1} dξ, keeping the solution u of (3.1) on K fixed. Since the solution on [h_t] is u_t and not u, the derivative also contains the interior term ∫_0^{ρ_K(v)} ∂_t[F^p(∇u_t(ξ v))] ξ^{n-1} dξ, which is neither computed nor bounded in the proof. A correct shape derivative must show that this term is absorbed through the equation for u_t and the boundary condition u_t=0 on ∂[h_t]; the present text does not supply such an argument. Because Lemmas 3.14, 4.3, 5.4, and 5.5 all invoke Lemma 3.13, Theorems 1.1, 1.2, and 1.3 rest on this missing step.","section":"§3.2, Lemma 3.13"},{"comment":"The existence arguments pass through Blaschke selection and obtain only a convex body K0 (or, in Lemma 5.13, a limit of polytopes), but the proof then invokes Lemma 3.8, which requires ∂K of class C^{2,α}, to obtain C^{1,α} solutions and uniform gradient bounds, and Lemma 3.11, which is stated only for C^{2,α} domains, to pass measures to the limit. The proof of Theorem 1.1 even says 'K0 ⊂ R^n be a bounded domain with the boundary of class C^{2,α}' although K0 is only known to be convex. Without an approximation argument, first by smooth bodies and then applying Lemmas 3.10–3.12 to the approximating sequence, the minimizer of Ψ_F,p,µ is not shown to be admissible for the variational formula, and the limiting measure identity in Lemma 5.13 is not justified. This gap affects both Theorem 1.1 and Theorem 1.3.","section":"§4, proof of Theorem 1.1; §5.2, Lemma 5.13"},{"comment":"Proposition 3.6(c) asserts that S_F,p(K,·) is translation invariant, and the proof derives this from the translation invariance of τ_F,p, which is logically insufficient: invariance of a scalar functional does not imply invariance of its associated measure. The translation invariance itself is true and can be proved directly from the change of variables y = x + z and the uniqueness of the solution, but as written it is unsupported. Lemma 4.1's centroid identity for S_F,p uses exactly this property, so the necessity part of Theorem 1.1 depends on this missing justification.","section":"§3.2, Proposition 3.6(c); §4, Lemma 4.1"}],"minor_comments":[{"comment":"The hypothesis 'such that ∂K up to set of (n−1)-dimensional Hausdorff measure zero' is incomplete; it should specify the required regularity of ∂K or state that the Gauss map is defined H^{n−1}-a.e., especially because the lemma is later applied to polytopes.","section":"§3.2, Lemma 3.13 statement"},{"comment":"In the proof of Lemma 3.11, the sentence 'from Lemma 3.11, since ui → u uniformly in C^1(\\bar K_i)' should cite Lemma 3.10 instead; also the convergence statement 'uniformly in C^1(\\bar K_i)' needs a fixed domain for the norm and should be formulated on a common compact set containing all K_i and K.","section":"§3.2, Lemma 3.11 proof"},{"comment":"Lemma 3.9 is essentially a consequence of the fact that u ∈ C^{1,α}(\\bar K), so the nontangential-limit statement follows from continuity; the proof as written is longer than necessary and the definition of Γβ(x) is not given.","section":"§3.2, Lemma 3.9"},{"comment":"There are numerous typos and minor notational slips: 'minximization' in the proof of Theorem 1.1, 'Porposition' at Lemma 2.1, 'track to' in the introduction, 'obatin' in Lemma 5.5, and some inconsistent use of 'v' versus 'u' for unit vectors. These do not affect the mathematics but should be corrected.","section":"Throughout"}],"recommendation":"major_revision","confidential_remarks":"The paper proposes a natural and potentially important extension of the Minkowski problem, and the variational strategy is appropriate. The main weakness is not the architecture of the proofs but the under-specified first-variation computation and the unjustified passage between smooth and non-smooth convex bodies. I believe both gaps are fixable within the scope of a revision, so I recommend major revision rather than rejection. The authors should also carefully compare their log-Minkowski results with the work of Hu (references [24] and [25]) and state precisely what is new in the anisotropic p-case."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The short version: the paper introduces genuinely new objects—the anisotropic p-torsional measure and its cone version—and states the right existence questions. If Theorem 1.1 is correct, it gives a complete characterization for the anisotropic p-torsional Minkowski problem, and Theorems 1.2–1.3 extend the log-Minkowski theory to this setting. The results recover Colesanti–Fimiani and Hu as special cases, and the overall strategy faithfully adapts known variational and approximation methods.\n\nHere is the problem. The proof of Lemma 3.13, the first variation formula, is not valid as written. It computes the derivative of τ_{F,p}([h_t]) by integrating the unperturbed solution u over the moving domain [h_t], and differentiates only the upper limit. But the solution on [h_t] is u_t, not u, so the interior derivative ∂_t F^p(∇u_t) is missing. That term can cancel through the PDE when you do the shape derivative properly, so the formula is probably true—but the paper never supplies that argument, and every subsequent variational step (Lemmas 3.14, 4.3, 5.4, 5.5) depends on Lemma 3.13. The stress-test note lands.\n\nThere is a second, related soft spot. After Blaschke selection in the proof of Theorem 1.1, the limit body is only known to be convex, but the argument invokes C^{2,α} gradient bounds from Lemma 3.8 and boundary gradient bounds. That step needs an approximation or stability argument that is not given.\n\nWhat the paper does well: the necessary conditions in Lemma 4.1 are clean, the functional setup is natural, and the discrete-to-continuous approximation for the log problem is standard and mostly careful. There are also typos—Lemma 3.10 states convergence in C^1(¯K_i) with different domains, Lemma 3.11's proof cites itself, and Lemma 4.1 has a garbled line. Those are minor and fixable.\n\nWho this is for: specialists in Minkowski-type problems for PDE-driven measures and people working on the Finsler p-Laplacian. A serious referee should see it; the potential is real. But I would not cite the theorems as established until Lemma 3.13 is repaired. My recommendation: send it out, but tell the referee to focus on the variational lemma first.","headline":"New objects and plausible theorems, but the proof of the key first-variation lemma freezes the PDE solution and that gap is load-bearing; worth refereeing if the author can repair it.","tokens_in":30404,"tokens_out":3211,"would_cite":false,"duration_ms":32920,"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":"Minkowski problem solved for anisotropic p-torsional rigidity","keywords":["Finsler p-Laplacian","anisotropic p-torsional rigidity","Minkowski problem","log-Minkowski problem","convex body","surface area measure","variational method","subspace mass inequality"],"falsifier":"Take a concrete convex body, for instance a cube in $\\mathbb{R}^2$ with $F$ the Euclidean norm, choose a continuous function $f$ on $S^1$, and compute the one-sided derivative of $\\tau_{F,p}([h_K+tf])$ at $t=0$ directly from the definition of the solution; if the result differs from $\\int f\\,dS_{F,p}(K,\\cdot)$, the variational reduction fails.","tokens_in":29363,"feed_emoji":"📐","tokens_out":5519,"duration_ms":50531,"temperature":0.7,"pith_summary":"This paper gives a complete existence characterization for the Minkowski problem in the anisotropic p-torsional setting. It shows that a nonzero finite Borel measure on the unit sphere is the anisotropic p-torsional measure of some convex body if and only if the measure is not concentrated on any closed hemisphere and its centroid is at the origin. The same variational route yields existence for the log-Minkowski problem of anisotropic p-torsional rigidity, first for discrete measures with support in general position and then, for $2 \\le p < \\infty$, for general measures satisfying a subspace mass inequality. This is the natural analogue of the classical torsional-rigidity Minkowski problem for the Finsler $p$-Laplacian, and it recovers the classical results when $p=2$ and $F$ is the sum-of-absolute-values norm.","feed_headline":"Minkowski problem solved for anisotropic p-torsional rigidity","feed_subtitle":"Two conditions on a measure decide if it is the torsional measure of some convex body.","key_machinery":"The central object is the anisotropic p-torsional measure $S_{F,p}(K,\\cdot)$, defined by integrating $F^p(\\nabla u)$ over the boundary portion with a given normal, where $u$ solves the Finsler $p$-Laplacian Dirichlet problem $\\Delta^F_p u=-1$ in $K$, $u=0$ on $\\partial K$. The argument is carried by the first-variation formula (Lemma 3.13): for support-function perturbations $h_t=h_K+t f$, the derivative of $\\tau_{F,p}([h_t])$ at $t=0$ equals $\\int f\\,dS_{F,p}(K,\\cdot)$. This identity converts the geometric Minkowski problem into a minimization problem for the functional $\\Psi_{F,p,\\mu}$, which is then solved by compactness: the measure conditions guarantee boundedness of the minimizing bodies, and continuity of $\\tau_{F,p}$ gives the limit. For the log problem, the same derivative appears in logarithmic form and controls the discrete maximization scheme.","core_discovery":"For $1<p<\\infty$, Theorem 1.1 characterizes the range of the map $K \\mapsto S_{F,p}(K,\\cdot)$: a nonzero finite Borel measure $\\mu$ on $S^{n-1}$ equals $S_{F,p}(K,\\cdot)$ for some convex body $K$ exactly when $\\mu$ is not concentrated on any closed hemisphere and $\\int_{S^{n-1}} v\\,d\\mu(v)=0$. The proof is variational: minimizing the functional $\\Psi_{F,p,\\mu}(K)=\\|h_K:\\mu\\|\\,/\\,\\tau_{F,p}(K)^{(p-1)/(np+p-n)}$ produces a convex body whose anisotropic p-torsional measure is proportional to $\\mu$, and a rescaling gives exact equality. Theorems 1.2 and 1.3 extend the same principle to the cone measure $\\tau^{\\log}_{F,p}$, proving existence for discrete measures in general position and, for $2\\le p<\\infty$, for measures satisfying the subspace mass inequality without any symmetry assumption.","pith_inferences":["The same variational scheme should apply to other functionals with a Pohozaev-type identity and a measure-valued first variation, such as anisotropic p-capacity, yielding analogous Minkowski-type existence theorems.","The p-dependent threshold in the subspace mass inequality may not be sharp; a natural test is whether the standard mass inequality $\\mu(\\xi\\cap S^{n-1})/|\\mu| < i/n$ already suffices for all $p\\in(1,\\infty)$, with the extra term an artifact of the proof.","The smooth case suggests a fully nonlinear Monge-Ampere equation of anisotropic type, so investigating $C^{2,\\alpha}$ regularity of the solution body for smooth densities is a direct next step.","A Gauss-curvature-flow approach could provide an alternative proof of Theorem 1.1 and might yield stability or uniqueness information that the variational argument does not address."],"forward_implications":["When $p=2$ and $F(\\xi)=\\sum_k |\\xi_k|$, Theorem 1.1 reduces to the Colesanti-Fimiani existence theorem for the classical torsional-rigidity Minkowski problem, and Theorem 1.3 recovers Hu's torsion log-Minkowski result.","If the measure has a smooth positive density $f$, the existence theorem is equivalent to solving the anisotropic Monge-Ampere equation $F^p(\\nabla u(g_K^{-1}(\\xi))) \\det(h_{ij}+h\\delta_{ij})(\\xi)=f(\\xi)$.","Every anisotropic p-torsional measure automatically has zero first moment, so the centroid condition is necessary and, by Theorem 1.1, sufficient together with the hemisphere condition.","For discrete log data in general position, the solution polytope has exactly $N$ facets, one for each atom of the measure.","Theorem 1.3 provides existence for the log-Minkowski problem of anisotropic p-torsional rigidity without evenness or symmetry assumptions, for $2\\le p<\\infty$."],"supporting_citations":[{"why":"Defines the anisotropic p-torsional rigidity and its variational characterization, which is the functional being varied throughout the paper.","marker":"[38]"},{"why":"Supplies the anisotropic Pohozaev identity and the boundary regularity of solutions used to derive the measure formulas and convergence lemmas.","marker":"[7]"},{"why":"Establishes the classical torsional-rigidity Minkowski problem, the baseline result that Theorem 1.1 extends and reduces to in the special case.","marker":"[17]"},{"why":"Provides the discrete log-Minkowski framework and the extremal functional used in the discrete existence proof of Theorem 1.2.","marker":"[47]"},{"why":"Contributes the subspace mass inequality technique and the compactness lemmas for polytopes that underlie the general log-Minkowski existence proof.","marker":"[22]"},{"why":"Solves the torsion log-Minkowski problem without symmetry assumptions, the classical counterpart that Theorem 1.3 generalizes to the anisotropic setting.","marker":"[25]"},{"why":"Gives the standard convex-geometry tools, including Blaschke selection and support-function properties, on which the variational compactness arguments rely.","marker":"[39]"}],"fun_headline_variants":["Minkowski problem cracked for anisotropic p-torsion","Anisotropic p-torsion Minkowski: full characterization","p-torsional Minkowski problem solved, log case too","Solving Minkowski for p-torsion: no symmetry needed","Minkowski problem for p-torsion: existence conditions found"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The argument depends on the first-variation formula for $\\tau_{F,p}$ under Wulff-shape perturbations; if that differentiation cannot be justified for all convex bodies in the variational argument, the main existence theorems lose their foundation.","fun_headline_variants_meta":{"raw":{"variants":["Minkowski problem cracked for anisotropic p-torsion","Anisotropic p-torsion Minkowski: full characterization","p-torsional Minkowski problem solved, log case too","Solving Minkowski for p-torsion: no symmetry needed","Minkowski problem for p-torsion: existence conditions found"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000445,"raw_usage":{"total_tokens":2198,"prompt_tokens":843,"completion_tokens":1355,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":459,"completion_tokens_details":{"reasoning_tokens":1266}},"tokens_in":459,"tokens_out":1355,"duration_ms":12130,"temperature":1.0,"reasoning_tokens":1266,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T22:46:16.039342+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take a concrete convex body, for instance a cube in $\\mathbb{R}^2$ with $F$ the Euclidean norm, choose a continuous function $f$ on $S^1$, and compute the one-sided derivative of $\\tau_{F,p}([h_K+tf])$ at $t=0$ directly from the definition of the solution; if the result differs from $\\int f\\,dS_{F,p}(K,\\cdot)$, the variational reduction fails.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Defines the anisotropic p-torsional rigidity and its variational characterization, which is the functional being varied throughout the paper."},{"cited_title":"Bianchin and G","cited_arxiv_id":null,"evidence_quote":"Supplies the anisotropic Pohozaev identity and the boundary regularity of solutions used to derive the measure formulas and convergence lemmas."},{"cited_title":"Colesanti and M.Fimiani, The Minkowski problem for torsional rigidity , Indiana Univ","cited_arxiv_id":null,"evidence_quote":"Establishes the classical torsional-rigidity Minkowski problem, the baseline result that Theorem 1.1 extends and reduces to in the special case."},{"cited_title":"Zhu, The logarithmic Minkowski problem for polytopes , Adv","cited_arxiv_id":null,"evidence_quote":"Provides the discrete log-Minkowski framework and the extremal functional used in the discrete existence proof of Theorem 1.2."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Contributes the subspace mass inequality technique and the compactness lemmas for polytopes that underlie the general log-Minkowski existence proof."},{"cited_title":"Schneider, Convex Bodies: The Brunn-Minkowski theory , 2nd edn, Cambridge Univ","cited_arxiv_id":null,"evidence_quote":"Gives the standard convex-geometry tools, including Blaschke selection and support-function properties, on which the variational compactness arguments rely."}],"review_version":1}