The Lq-Minkowski problem of anisotropic p-torsional rigidity

Bin Chen; Chao Li

REVIEW 4 major objections 4 minor 42 references

The Lq-Minkowski problem of anisotropic p-torsional rigidity

T0 review · 4 major / 4 minor · reviewed 2026-08-08 · deepseek-v4-flash

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Pith's one-line read 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.

desk verdict 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. read the letter →

arxiv 2502.05804 v2 pith:NBPKMWB7 submitted 2025-02-09 math.AP

Chao Li , Bin Chen This is my paper · ORCID

classification math.AP MSC 35N2552A2053C2131A15

keywords Lq-Minkowskiproblemanisotropicp-torsionalrigidityp-LaplacianconvexbodiesWulffshapeBorelmeasurevariationalmethod

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 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

What carries the argument

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.

What would settle it

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.

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Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

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.

Reading between the lines

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

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.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Colleague,

Read 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

The 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.

There 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.

That 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.

For 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.

Would 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.

Referee Report

4 major / 4 minor

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.

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 (4)

[§2.2, Lemma 2.4] 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.
[§1 and §2.2, definition of S_{F,p,q}] 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.
[§4, Lemma 4.3] 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.
[§3, Theorem 3.2] 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.

minor comments (4)

[§2.2, Lemma 2.4] 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.
[§1] There is a typo in 'Meanehile' in the introduction; it should be 'Meanwhile'.
[§2.1] The reference to 'Porposition 2.5' in Lemma 2.1 should read 'Proposition 2.5'.
[§2.2, equation (2.15)] 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).

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the q≠1 existence theorems are not forced by the definitions or by the self-cited q=1 results; the variational lemma is an in-paper derivation attempt, whose proof gap is a correctness issue, not a circular reduction.

full rationale

I walked the derivation chain. The Lq anisotropic p-torsional measure S_{F,p,q}(K,·) is defined independently as an integral over the boundary (intro and (2.15)), and the main existence theorems (Theorems 1.1 and 1.3) are proved by variational minimization and polyhedral approximation. The only in-paper derivation that could make the conclusion an input is Lemma 2.4, but its statement is a formula for the variation of τ_{F,p} in terms of S_{F,p,q}; it is not used to define S_{F,p,q} in a way that presupposes the target measure. The proof does contain a gap: it differentiates the moving boundary ρ[h_t] while keeping the PDE solution u fixed, which is a correctness concern, not a circularity. The self-citations to Li [29] supply the q=1 anisotropic p-torsional measure and properties such as translation invariance, homogeneity, and continuity; these are background facts and do not contain the q≠1 existence claims, nor are they fitted parameters. The minimization problems involve no parameters fitted to the target measure, and the final equations are Euler-Lagrange identities derived from the variational formulas, not renaming of known results. Minor internal inconsistency: the factor (p−1)/(n(p−1)+p) appears in the intro definition of S_{F,p,q} but not in (2.15); this affects normalization but not circularity. Overall score 1 reflects only the presence of a same-author citation in the background, not load-bearing circularity.

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

No fitted numerical parameters appear in the paper. The proof relies on imported regularity, continuity, and variational results from the author's earlier q=1 paper [29] and from [9,25]. The Lq anisotropic p-torsional measure is a new definition, not an unexplained postulated entity.

assumptions (5)

domain assumption F is a regular norm in I_p = {F∈C^{2,α}(R^n\{0}): (1/p)F^p∈C^2_+(R^n\{0})} and is strictly convex.
Regularity of the anisotropic p-Laplacian solution and the measure computations in Section 2.2 depend on this; it is assumed without proof.
domain assumption Anisotropic Pohozaev identity and integral expression τ_{F,p}(K)= (p-1)/(n(p-1)+p)∫_{S^{n-1}} h_K F^p(∇u) dS.
Imported from [2,29]; this formula defines the torsional measure and is used throughout.
domain assumption Continuity, homogeneity, translation invariance, and monotonicity of τ_{F,p} and S_{F,p} in the Hausdorff metric.
Taken from [29]; used in Blaschke selection and approximation arguments in Sections 3 and 4.
domain assumption Upper bound τ_{F,p}(K) ≤ (p-1)/(n(p-1)+p) n^{-1/(p-1)} κ_n^{-p/(n(p-1))} V(K)^{(n(p-1)+p)/(n(p-1))}.
Lemma 2.3 from [9]; used to show that limit bodies have positive volume.
domain assumption For 0<q<1, Φ_{f,μ}(ξ)=∫(f(u)-ξ·u)^q dμ(u) is strictly concave and has a unique maximizer ξ_f depending continuously on f.
Lemma 4.1 from [25]; the core of the polyhedral approximation proof for 0<q<1.

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Pith. "Pith review of The Lq-Minkowski problem of anisotropic p-torsional rigidity." pith.science (2026). https://pith.science/paper/NBPKMWB7

@misc{pith2026250205804,
  author       = {Pith},
  title        = {Pith review of: The Lq-Minkowski problem of anisotropic p-torsional rigidity},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/NBPKMWB7}},
  note         = {Machine review of arXiv:2502.05804}
}

abstract

In this paper, the $L_q$-Minkowski problem of anisotropic $p$-torsional rigidity is considered. The existence of the solution of the $L_q$-Minkowski problem of anisotropic $p$-torsional rigidity with $0<q<1$ and $1<q\neq \frac{p}{p-1}+n$ is given.

Discussion (0). Continue with ORCID to comment.

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Pith tools

Reviewed August 8, 2026 · model on record in the stance chip above.

[1] [29]

Minkowski problem of anisotropic p-torsional rigidity

C. Li, Minkowski problem of anisotropic p-torsional rigidity , arXiv:2501.00687v2, 2025. 2, 5

work page Pith review arXiv 2025

[2] [9]

Della Pietra and N

F. Della Pietra and N. Gavitone, Sharp bounds for the ﬁrst eigenvalue and the torsional rigid ity related to some anisotropic operators , Math. Nachr., 287 (2014), 194-209. 1, 2, 5

work page 2014

[3] [1]

Akman and S

M. Akman and S. Mukherjee, On the Minkowski problem for p-harmonic measures, Calc. Var. Partial Diﬀerential Equations, 64 (2025), 1: 36. 1

work page 2025

[4] [2]

Bianchin and G

C. Bianchin and G. Ciraolo, Wulﬀ shape characterizations in overdetermined anisotrop ic elliptic problems, Comm. Partial Diﬀerential Equations, 43 (2018), 5: 790-820. 2

work page 2018

[5] [3]

K. J. B¨ or¨ oczky, E. Lutwak, D. Yang and G. Zhang, The Gauss image problem , Comm. Pure Appl. Math., 73 (2020), 7: 1406-1452. 1

work page 2020

[6] [4]

Chen, C Li and W Wang, The Lp Gauss dual Minkowski problem , arXiv:2412.13557, 2024

B. Chen, C Li and W Wang, The Lp Gauss dual Minkowski problem , arXiv:2412.13557, 2024. 1

work page arXiv 2024

[7] [5]

B. Chen, X. Zhao, W. Wang and P. Zhao, The Lp Minkowski problem for q-torsional rigidity , Adv. Calc. Var., 17 (2024), 3: 587-603. 2, 3 18 CHAO LI AND BIN CHEN

work page 2024

[8] [6]

Chen and Q

Z. Chen and Q. Dai, The Lp Minkowski problem for torsion , J. Math. Anal. Appl., 488, (2020), DOI: 10.1016/j.jmaa.2020.124060. 2, 3

work page arXiv 2020