Uniqueness of solutions to the isotropic $L_{p}$ Gaussian Minkowski problem

Jinrong Hu

Uniqueness of solutions to the isotropic $L_{p}$ Gaussian Minkowski problem

T0 review · 0 major / 4 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read For -n-1 < p < -1, every admissible solution inside the unit ball is a sphere.

desk verdict A clean, checkable uniqueness proof for the Lp Gaussian Minkowski problem in a narrow but genuinely new range; worth refereeing, with the R(K)<=1 caveat kept visible. read the letter →

arxiv 2412.12851 v2 pith:2OH665RD submitted 2024-12-17 math.AP math.MG

Jinrong Hu This is my paper · ORCID

classification math.APmath.MG MSC 35A0252A20

keywords uniquenessisotropicL_pGaussianMinkowskiproblemsurfaceareameasureMonge-AmpèreequationconvexbodieslocalBrunn-Minkowskiinequalitysupportfunction

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The pith

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

The reading

The paper proves a uniqueness theorem for the isotropic $L_p$ Gaussian Minkowski problem in every dimension $n \ge 1$, in the negative range $-(n+1) < p < -1$. The equation studied is the Monge-Ampère type equation $h^{1-p}e^{-|Dh|^2/2}/\kappa = c$ on the sphere, which says that the $L_p$ Gaussian surface area measure of the body is uniform. The theorem removes the usual assumption that the body is centered at the origin: any smooth strictly convex solution with support function $h>0$ and with the body lying in the unit ball ($R(K) \le 1$) must be a sphere. This matters because uniqueness of such solutions is what makes degree-theoretic existence proofs for the non-normalized problem well defined, so the result opens the way to existence of small solutions in a nonsymmetric setting. The proof uses a spectral formulation of the Alexandrov-Fenchel inequality with Gaussian-weighted test functions, and the unit-ball condition supplies the final sign.

What carries the argument

The argument is carried by a spectral formulation of the Alexandrov-Fenchel inequality (Lemma 3.1): for functions with zero mean against $h\sigma_k$, one has $k\int_{S^n} f^2 h\sigma_k\,d\sigma \le \int_{S^n} h^2\sigma^{ij}_k\nabla_i f\nabla_j f\,d\sigma$, with equality only for $f=\langle x/h,v\rangle$. The proof feeds in the Gaussian-weighted coordinate functions $f_l=e^{-|X|^\alpha/2}\langle X,E_l\rangle$ minus their means, sums over an orthonormal basis, and obtains the weighted inequality of Lemma 3.2. Substituting the equation in the form $h^{n+2}/\kappa=ce^{|X|^\alpha/2}h^{n+1+p}$ converts the main term into $(n+1+p)\int |\nabla h|^2 e^{-|X|^\alpha}\,dV_n$ plus a leftover term, and the condition $R(K)\le1$ makes that leftover term carry the sign $(|X|^2-1)\le0$. The chain forces $|\nabla h|\equiv0$, so $h$ is constant and $\partial K$ is a sphere.

What would settle it

Compute the linearized isotropic $L_p$ Gaussian Minkowski equation around a centered ball of radius $r<1$ in dimension $n=1$ or $n=2$. The theorem predicts that every nonconstant mode vanishes for $-(n+1)<p<-1$; exhibiting a nonzero mode, or producing a non-spherical solution with $R(K)\le1$ for some $c>e^{-1/2}$, would falsify the claim.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central discovery is Theorem 1.1: let $n\ge1$ and $-(n+1)<p<-1$. If $\partial K$ is a smooth, strictly convex hypersurface with support function $h>0$ and $R(K)\le1$, and if $h^{1-p}e^{-|Dh|^2/2}/\kappa=c$ for some constant $c>0$ on $S^n$, then $\partial K$ is a sphere. Consequently, for $c\in(0,e^{-1/2}]$ the unique solution is the centered ball of radius $r\in(0,1]$ solving $r^{n+1-p}e^{-r^2/2}=c$, while for $c>e^{-1/2}$ there is no solution at all in this class. The advance over prior work is that the origin-centred assumption on the body is dropped, at the price of the size condition $R(K)\le1$.

Load-bearing premise

The load-bearing premise is the size bound $R(K)\le1$, meaning every point of the convex body is within distance 1 of the origin. At equation (3.11) this makes $(|X|^2-1)\le0$, which supplies the sign that forces $|\nabla h|=0$; without the bound, centered balls of radius larger than 1 can both satisfy the equation, so uniqueness fails.

Editorial extensions

If this is right

For every $n\ge1$ and every $p\in(-(n+1),-1)$, the only smooth strictly convex solution of the isotropic $L_p$ Gaussian Minkowski problem with $R(K)\le1$ is a centered sphere.
The origin-centred assumption, previously required for $p>-n-1$, is not needed in this range once the body lies in the unit ball.
For $0<c\le e^{-1/2}$, the unique solution is the centered ball of radius $r\in(0,1]$ with $r^{n+1-p}e^{-r^2/2}=c$; for $c>e^{-1/2}$, the theorem implies there is no solution at all in this class.
The uniqueness statement is what makes the degree-theoretic construction of small solutions to the non-normalized $L_p$ Gaussian Minkowski problem well defined for $-(n+1)<p<-1$.
The obstruction to larger bodies is real: without $R(K)\le1$, centered balls of radius greater than 1 provide non-unique solutions, so any extension needs a new inequality.

Reading between the lines

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

Editorial inference: the same test-function scheme should transpose to other radial densities $e^{-\varphi(|X|)}$, with the unit-ball bound replaced by a monotonicity condition on $\varphi$, since the decisive sign in the proof comes from the derivative of the Gaussian exponent.
Editorial inference: at the endpoint $p=-1$ the coefficient $(n+1+p)(-1-p)$ in the main estimate vanishes, so this argument degenerates and cannot by itself decide the logarithmic Gaussian case.
Editorial inference: the appendix's local spectral Ehrhard inequality suggests that a suitably chosen test function could replace $R(K)\le1$ by a Gaussian-volume condition such as $\gamma(K)\ge1/2$, extending uniqueness beyond the unit ball.

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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

Short version: the theorem is new as stated, and the proof is sound as far as I can tell. I rechecked the main chain—Lemmas 3.2, 3.3, and the substitution into Theorem 1.1—and did not find a missing sign or a hidden circular step. The stress-test note holds up: R(K)<=1 is used exactly where (3.11) needs (|X|^2-1)<=0, and it is stated explicitly in the theorem. This is a real extension, since Ivaki-Milman proved uniqueness for p>-n-1 only in the origin-centered class, and the planar results cover p>=0. The new test function with Gaussian weight and variance subtraction is the right idea, and the algebra using the sigma_1 sigma_n and sigma_n identities checks out.

Credit where due: the paper does not oversell. The limitation of the spectral method for R(K)>1 is acknowledged in the introduction and again in Remark A.4, and the constant-solution count is correct under R<=1 because g(t)=t^{n+1-p}e^{-t^2/2} is increasing on (0,1] when p is in (-n-1,-1). I would ask for two small changes: put R(K)<=1 in the abstract—the current abstract says "without requiring the origin-centred assumption" and omits the size bound, which is misleading in isolation—and spell out the centered-ball example showing non-uniqueness for R(K)>1 in one sentence. The r->0 and r->infinity asymptotic in the introduction already hints at this, but the explicit two-ball statement would help.

The soft spots are minor. Lemma 3.1 is a standard spectral Alexandrov-Fenchel inequality quoted without proof; that is fine for a specialist journal, though a more precise pointer to the exact theorem in [3] or [5] would help. Appendix A is speculative and unused; it does not hurt, but it could be trimmed or more clearly labeled as a remark. The citation pattern is standard, and no self-cited result is load-bearing.

Who this is for: convex geometers working on Minkowski-type problems and anyone using degree-theoretic existence in Gaussian settings. This is a meaningful step, not a watershed, but it deserves a serious referee. I expect acceptance after the abstract and remark fixes.

Referee Report

0 major / 4 minor

Summary. The paper establishes a uniqueness theorem for the isotropic L_p Gaussian Minkowski problem in R^{n+1}. Theorem 1.1 states that for -(n+1)<p<-1, any smooth strictly convex hypersurface with support function h>0 and R(K)<=1 satisfying h^{1-p} e^{-|Dh|^2/2} (1/kappa) = c for c>0 must be a sphere. The proof uses the spectral formulation of the Alexandrov-Fenchel inequality (Lemma 3.1) with carefully chosen test functions depending on the position vector X=Dh, derives a differential inequality via Lemmas 3.2 and 3.3, and then uses the size condition R(K)<=1 to force |grad h|=0. The paper also discusses constant solutions and includes an appendix on a local Ehrhard inequality.

Significance. If correct, the result is a genuine advance: it removes the origin-centred assumption for p<1 in the supercritical range -(n+1)<p<-1, under the explicit and honestly stated size condition R(K)<=1. The proof is coherent and self-contained modulo the standard spectral Alexandrov-Fenchel inequality, and the algebraic identities in Lemmas 3.2 and 3.3 check out, including the sign in Eq. (3.11) and the use of R(K)<=1. The paper also correctly identifies the obstruction to the method for R(K)>1 by exhibiting non-uniqueness among centered balls of different radii. These are concrete strengths. The main limitation is that the theorem is not as general as the abstract suggests, since the R(K)<=1 hypothesis is omitted there.

minor comments (4)

[Abstract] The abstract states uniqueness in the range -(n+1)<p<-1 without mentioning the hypothesis R(K)<=1. As written, this overclaims: without that condition the statement is false, since centered balls of different radii larger than 1 can satisfy the same equation. The abstract should include the size restriction.
[Theorem 1.1] The final sentence of Theorem 1.1 says that for c>e^{-1/2} there is no constant solution. This is only true under the standing assumption R(K)<=1; without it, the function g(t)=t^{n+1-p}e^{-t^2/2} has values above e^{-1/2} on (1, sqrt(n+1-p)), giving two constant solutions with R(K)>1. Please add an explicit qualifier such as 'under the condition R(K)<=1' to this sentence.
[References] There are several typographical errors in the references: [5] contains 'and and', [26] contains 'Gauassian', and [33] contains 'to apprear'. The spelling of Alexandrov/Aleksandrov is inconsistent between Lemma 3.1 and references [1,2].
[Appendix A] The notation in the proof of Lemma A.2, especially the terms involving 'h_i|Dh|_j|Dh|' in Eq. (A.3), is very difficult to parse. Since this appendix is not used in the proof of the main theorem, consider rewriting it more carefully or clearly marking it as heuristic. This does not affect the main result.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the proof derives uniqueness from an external spectral Alexandrov–Fenchel inequality and the PDE itself, with no fitted parameters or load-bearing self-citations.

full rationale

The derivation chain is self-contained against external results. Lemma 3.1 is quoted from Andrews [3,5], and the test functions in Lemma 3.2 follow the method of Ivaki-Milman [23], an independent source. The proof of Theorem 1.1 substitutes the PDE relation h^{1-p} e^{-|Dh|^2/2} (1/κ) = c into Lemma 3.3, applies Cauchy-Schwarz and the hypothesis R(K) ≤ 1, and obtains an inequality whose left-hand side is nonnegative while the right-hand side is nonpositive, forcing |∇h| = 0. No parameter is fitted and no 'prediction' is derived from data or from a prior claim of the same paper. The only self-citation, [18], appears in the introduction as contextual literature and plays no role in the proof. The constant-solution count for centered balls is an independent check and is not part of the uniqueness argument. The restriction R(K) ≤ 1 is an explicit hypothesis and is used exactly where stated; its limitation is acknowledged in the text and does not constitute circularity.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

No free parameters are fitted: p is the fixed exponent in the problem statement and c is the prescribed constant. No new entities are introduced; the proof uses only the support function h, the Gauss curvature κ, and the standard inequality from [3,5].

assumptions (3)

standard math Spectral Alexandrov-Fenchel inequality (Lemma 3.1) for f with zero weighted mean and its equality case
Stated in Section 3 and used in Lemma 3.2 to produce inequality (3.1); it is cited to Andrews [3] and Andrews et al. [5] and not proved in this paper.
domain assumption Smoothness and strict convexity of ∂K, h>0
Assumed in Theorem 1.1 so the support function, Gauss map, principal radii, and the Monge-Ampere equation are classical; also ensures dV_n = hσ_n dσ is a positive measure.
domain assumption Size restriction R(K)≤1
Assumed in Theorem 1.1 and used after Eq. (3.11): with α=2 the right-hand side contains (|X|^2-1), and R(K)≤1 makes this nonpositive, forcing |∇h|=0. The author notes the method does not work for R(K)>1.

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Pith. "Pith review of Uniqueness of solutions to the isotropic $L_{p}$ Gaussian Minkowski problem." pith.science (2026). https://pith.science/paper/2OH665RD

@misc{pith2026241212851,
  author       = {Pith},
  title        = {Pith review of: Uniqueness of solutions to the isotropic $L_p$ Gaussian Minkowski problem},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2OH665RD}},
  note         = {Machine review of arXiv:2412.12851}
}

abstract

The uniqueness of solutions to the isotropic $L_{p}$ Gaussian Minkowski problem in $\mathbb{R}^{n+1}$ is established when $-(n+1)<p<-1$ with $n\geq 1$, without requiring the origin-centred assumption on convex bodies.

Discussion (0). Continue with ORCID to comment.

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Works this paper leans on

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

[1] [3]

Andrews, Monotone quantities and unique limits for ev olving convex hypersurfaces, Internat

B. Andrews, Monotone quantities and unique limits for ev olving convex hypersurfaces, Internat. Math. Res. Notices 1997, no. 20, 1001–1031

work page 1997

[2] [5]

Andrews, B

B. Andrews, B. Chow, C. Guenther and and M. Langford, Extrinsic geometric ﬂows , Graduate Studies in Mathematics, 206, Amer. Math. Soc., Providence, RI, [202 0] ©2020

work page 2020

[3] [1]

A. D. Aleksandrov, On the surface area measure of convex b odies, Mat. Sb. (N.S.) 6 (1939), 167–174

work page 1939

[4] [2]

A. D. Aleksandrov, On the theory of mixed volumes. III. Ex tensions of two theorems of Minkowski on convex polyhedra to arbitrary convex bodies, Mat. Sb. 3 (1938), 27–46

work page 1938

[5] [4]

Andrews, Classiﬁcation of limiting shapes for isotro pic curve ﬂows, J

B. Andrews, Classiﬁcation of limiting shapes for isotro pic curve ﬂows, J. Amer. Math. Soc. 16 (2003), no. 2, 443–459. 12

work page 2003

[6] [6]

Bianchi, K

G. Bianchi, K. J. B¨ or¨ oczky, A. Colesanti and D. Yang, Th e Lp-Minkowski problem for − n < p < 1, Adv. Math. 341 (2019), 493–535

work page 2019

[7] [7]

Bonnesen and W

T. Bonnesen and W. Fenchel, Theory of convex bodies , translated from the German and edited by L. Boron, C. Christenson and B. Smith, BCS Associates, Moscow, ID, 1987

work page 1987

[8] [8]

B¨ or¨ oczky, E

K. B¨ or¨ oczky, E. Lutwak, D. Yang and G. Zhang, The logarithmic Minkowski problem, J. Amer. Math. Soc. 26 (2013), no. 3, 831–852

work page 2013