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Minkowski problem of anisotropic p-torsional rigidity

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Pith's one-line read Minkowski problem solved for anisotropic p-torsional rigidity

desk verdict 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. read the letter →

arxiv 2501.00687 v3 pith:NBGRUVJG submitted 2025-01-01 math.AP math.DG

Chao Li This is my paper · ORCID
classification math.APmath.DG MSC 35N2552A2053C2131A15
keywords Finslerp-Laplaciananisotropicp-torsionalrigidityMinkowskiproblemlog-Minkowskiconvexbodysurfaceareameasurevariationalmethodsubspacemassinequality
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

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.

What carries the argument

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.

What would settle it

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.

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Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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 $2\le p<\infty$.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • 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.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 4 minor

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.

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 (3)
  1. [§3.2, Lemma 3.13] 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.
  2. [§4, proof of Theorem 1.1; §5.2, Lemma 5.13] 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.
  3. [§3.2, Proposition 3.6(c); §4, Lemma 4.1] 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.
minor comments (4)
  1. [§3.2, Lemma 3.13 statement] 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.
  2. [§3.2, Lemma 3.11 proof] 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.
  3. [§3.2, Lemma 3.9] 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.
  4. [Throughout] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity found: the anisotropic p-torsional measure is defined independently, the main existence proofs follow a standard variational route from the stated first-variation formula, and the only self-citation is historical and not load-bearing.

full rationale

The paper's central claims are not circular. The anisotropic p-torsional measure S_{F,p}(K,·) is defined in Section 3.1 by equation (3.10) as an integral of F^p(∇u) over the inverse Gauss image, and Theorem 1.1 is proved by minimizing Ψ_{F,p,µ}(K) = ∫ h_K dµ / τ_{F,p}(K)^{(p-1)/(np+p-n)}. The Euler-Lagrange step uses the first variation formula of Lemma 3.13, which states that the derivative of τ_{F,p}([h_t]) at t=0 is ∫ f dS_{F,p}(K,·). This formula connects the variation to the measure, but it is not circular: the measure S_{F,p} is not fitted to any data, and no parameter is chosen so that the formula holds. The proof of Lemma 3.13 is indeed mathematically questionable: it differentiates the moving Wulff shape [h_t] while keeping the PDE solution u fixed and omits the t-derivative of the solution u_t. That is a correctness gap, not a circularity, because the omitted interior term is not the target measure. The paper also relies on external, independently stated results: [7] for C^{1,α} regularity and gradient bounds, [38] for the variational characterization of τ_{F,p}, [17] for convergence of boundary integrals, and [47,22] for the discrete log-Minkowski framework. The only self-citation, [33] by Li and Zhao, appears solely in the historical introduction and is not used in any proof. The necessity conditions in Theorem 1.1 are verified from translation invariance and the variational formula; sufficiency follows from compactness, continuity, and the same formula. The subspace mass inequality in Theorem 1.3 is an explicit hypothesis rather than a conclusion smuggled in. No quoted step reduces by construction to its own input, so the circularity score is 0.

Assumptions & free parameters 0 free parameters · 5 assumptions · 2 invented entities

The central claim rests on standard convex geometry facts, regularity theory for the Finsler p-Laplacian, and the anisotropic Pohozaev identity, all drawn from the cited literature. No numerical free parameters are fitted. The subspace mass inequality in Theorem 1.3 is an explicit hypothesis, not an axiom.

assumptions (5)
  • domain assumption F is a regular norm in I_p (C^{2,α}(R^n\{0}) and (1/p)F^p has positive definite Hessian)
    Invoked in Lemma 3.8 to get C^{1,α} regularity and the boundary gradient bounds (3.13).
  • standard math Anisotropic Pohozaev identity (3.8) from Bianchin-Ciraolo [7]
    Used to derive formula (3.9) connecting τ to the boundary integral and to check centroid conditions.
  • standard math Blaschke selection theorem and the stated continuity of τ_{F,p} under Hausdorff convergence
    Used to extract convergent minimizing sequences in the proofs of Theorem 1.1, Lemma 5.4, and Lemma 5.13.
  • standard math John ellipsoid containment E_k⊂P̄_k⊂n(E_k-o_k)+o_k
    Used in Lemma 5.12 to bound the approximating polytopes.
  • standard math Wulff shape properties, including h_{[h]}≤h and the radial derivative formula of Lemma 2.2
    Used in the variational arguments of Lemma 3.13 and Lemma 4.3.
invented entities (2)
  • Anisotropic p-torsional measure S_{F,p}(K,·)
    purpose: The boundary measure prescribed in the Minkowski problem; defined in (3.10).
    A mathematical definition built from the Finsler p-Laplacian solution; no external falsifiable handle.
  • Cone anisotropic p-torsional measure τ^log_{F,p}(K,·)
    purpose: The measure for the logarithmic Minkowski problem; defined in (3.11).
    A mathematical definition built from S_{F,p} weighted by the support function; no external falsifiable handle.

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Pith. "Pith review of Minkowski problem of anisotropic p-torsional rigidity." pith.science (2026). https://pith.science/paper/NBGRUVJG

@misc{pith2026250100687,
  author       = {Pith},
  title        = {Pith review of: Minkowski problem of anisotropic p-torsional rigidity},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/NBGRUVJG}},
  note         = {Machine review of arXiv:2501.00687}
}
abstract

In this paper, we consider the Minkowski problem associated with the solution to the anisotropic $p$-Laplacian (or Finsler $p$-Laplacian) equation, namely, the Minkowski problem of anisotropic $p$-torsional rigidity. The sufficient and necessary conditions for the existence of a solution to the Minkowski problem of anisotropic $p$-torsional rigidity are presented. Meanwhile, the existence of a solution to the log-Minkowski problem of anisotropic $p$-torsional rigidity without symmetry assumptions is solved.

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. The Lq-Minkowski problem of anisotropic p-torsional rigidity

    math.AP 2025-02 reject novelty 6.0 of 10

    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.

Reference graph

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