The even Lp Gaussian dual Minkowski problem

J.C.Liu; W.Shi

REVIEW 4 major objections 4 minor 66 references

The even Lp Gaussian dual Minkowski problem

T0 review · 4 major / 4 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read 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.

desk verdict 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. read the letter →

arxiv 2412.13651 v1 pith:4OQDMCAD submitted 2024-12-18 math.FA

W.Shi , J.C.Liu This is my paper · ORCID

classification math.FA MSC 52A4052A3835J96

keywords LpGaussiandualcurvaturemeasureMinkowskiproblemMonge-AmpereequationconvexbodyevendensityvariationalmethodWulffshapequermassintegral

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 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.

What carries the argument

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.

What would settle it

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.

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Extended reading notes

Core claim

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 $S^{n-1}$.

Load-bearing premise

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.

Editorial extensions

If this is right

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 $\tilde C_{p,\gamma_{n,q}}(K,\cdot)$.

Reading between the lines

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

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.

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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

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.

Second, 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.

Also, 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.

Credit 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.

For 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.

Referee Report

4 major / 4 minor

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.

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 (4)

[§4.2, Theorem 4.2 vs. Theorem 1.1] 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.
[§4.2, lower bound for h_{K_i}] 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.
[§4.1, Theorem 4.1, equation (4.5)] 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.
[§4.3, Theorem 4.3] 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.

minor comments (4)

[Abstract] 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.
[§3, end of section] 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.
[§4.1, notation] 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.
[§3, Lemma 3.1] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the proof is a genuine variational argument, and the mismatch between the hypotheses of Theorem 1.1 and Theorem 4.2 is a derivation gap, not a circular reduction.

full rationale

The paper's derivation chain is not circular. The Lp Gaussian dual curvature measure is introduced through a variational formula proved in Theorem 3.1, using Lemma 3.1 from [40] and Lemma 3.2 from [18]; these are published technical lemmas that do not assume the target existence result. The optimization problem in (4.3) and the Euler-Lagrange argument in Theorem 4.1 are standard: a constrained maximizer is shown to satisfy the normalized measure equation, and no fitted parameter or target quantity is smuggled into the input. Theorem 4.2 establishes existence of the maximizer under the extra assumption 1/M0 ≤ g(u) ≤ M0, and this assumption is absent from the statement of Theorem 1.1. That is a genuine correctness gap in the proof as written, but it is not a circular step: the theorem is not reduced to its conclusion by definition, and the missing lower bound is not a renamed version of the desired result. The self-citation [40] by co-author Liu supplies a radial-derivative formula, but the central existence claim does not reduce to that citation. No circular step can be exhibited by quoting an equation where the output equals the input by construction.

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

No fitted parameters appear. The proof depends on standard convex-geometry tools, two cited lemmas from prior work, a bounded-density assumption omitted from the main theorem, and an unproved Lagrange multiplier assertion. The Lp Gaussian dual curvature measure is a defined mathematical object rather than an invented physical entity requiring independent evidence.

assumptions (5)

domain assumption Lemma 3.1 from [40]: Lp radial growth formula (3.2) and uniform Lipschitz bound (3.3) for Wulff shapes of h_t=(h_K^p+tf^p)^{1/p}.
Quoted from the authors' prior paper and used to derive Theorem 3.1 and the optimization derivative; not proved here.
domain assumption Lemma 3.2 from [18]: continuity of the Gaussian dual quermassintegral under Hausdorff convergence for q>1.
Used to pass limits in Theorem 4.2; cited without proof.
domain assumption Uniform lower bound min_v h_{K_i}(v) ≥ M1 under the constraint ilde V_{γ_{n,q}}(K_i)=|μ| and bounded density, as stated in [18].
Invoked in Theorem 4.2 to get compactness; not proved in this paper.
ad hoc to paper The prescribed measure has density g with 1/M0 ≤ g(u) ≤ M0 for some M0>0.
Used to establish upper and lower bounds on support functions in Theorem 4.2 but absent from Theorem 1.1.
ad hoc to paper Existence of a Lagrange multiplier λ for the constrained maximization in Theorem 4.1.
Implicitly assumed when differentiating Γ(t,λ) and setting the derivative to zero; not justified, especially because the perturbation h_t is not feasible.

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Pith. "Pith review of The even Lp Gaussian dual Minkowski problem." pith.science (2026). https://pith.science/paper/4OQDMCAD

@misc{pith2026241213651,
  author       = {Pith},
  title        = {Pith review of: The even Lp Gaussian dual Minkowski problem},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/4OQDMCAD}},
  note         = {Machine review of arXiv:2412.13651}
}

abstract

The even Gaussian dual Minkowski problem studied by Feng, Hu and Xu, In this paper, we consider the even $L_p$ dual-Gaussian Minkowski problem for $p>1$. The existence of $o$-symmetric solution in the case $p>1$ is obtained.

Discussion (0). Continue with ORCID to comment.

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Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.

[1] [18]

Y. Feng, S. Hu, and L. Xu. Existence of solutions to the ev en Gaussian dual Minkowski problem. Advances in Applied Mathematics , 163:102808, 2025

work page 2025

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