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Uniqueness in the near isotropic Lp dual Minkowski problem

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Pith's one-line read For p in (-1,1), near-isotropic solutions of the Lp dual Minkowski problem are unique.

desk verdict The p∈(-1,0) uniqueness result is promising but the proof's C^0 estimate has a real gap: the key contradiction only works for p=0. read the letter →

arxiv 2505.01066 v2 pith:2OLRSBCO submitted 2025-05-02 math.AP

Karoly J. Boroczky , Shibing Chen , Weiru Liu , Christos Saroglou This is my paper · ORCID
classification math.AP MSC 52A2035J96
keywords LpdualMinkowskiproblemnear-isotropicuniquenessC0estimateconvexbodysurfaceareameasurecurvatureweaksolution
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

The paper proves that the near-isotropic $L_p$ dual Minkowski problem has a unique solution: if the exponent $p$ lies between $-1$ and $1$, the dual exponent $q$ is sufficiently close to the space dimension $n$, and the prescribed density $f$ on the sphere is Hölder-close to the constant function $1$, then exactly one convex body (in the weak measure sense) has that density as its $L_p$ dual curvature measure. This closes the previously open negative range $-1

What carries the argument

The load-bearing object is the $L_p$ $q$th dual curvature measure $\tilde C_{p,q,K}$ on the sphere, defined by $d\tilde C_{p,q,K}=h_K^{-p}\,d\tilde C_{q,K}$, where $\tilde C_{q,K}(\omega)=\int_{\alpha_K^*(\omega)}\rho_K^q\,dH^{n-1}$ records how much of the boundary of $K$ is seen radially from directions in $\omega$. The argument runs on two mechanisms. First, a $C^0$ estimate (Theorem 1.6) is proved by anisotropic blow-up: if a body with nearly uniform $\tilde C_{p,q,K}$ became unbounded, an affine rescaling would force its $L_p$ surface-area measure to converge to a measure supported on a lower-dimensional subspace with constant density there, and a non-existence result (Theorem 4.5) rules that configuration out; the same estimate also rules out collapse to zero volume. Second, an inverse-function step at the constant solution uses the linearized operator $L\varphi=\Delta_{S^{n-1}}\varphi+(n-p)\varphi$, which is invertible on Hölder spaces because the spherical Laplacian has eigenvalues $k(k+n-2)$ and because $n-p>0$ for $p<1$. The $C^0$ estimate upgrades this local diffeomorphism into a global uniqueness theorem.

What would settle it

Fix $n=2$, $p\in(-1,0)$, $q=2$, and take $f_\varepsilon=1+\varepsilon\,Y_{1,0}$ on $S^1$, with $Y_{1,0}$ the first spherical harmonic and $\varepsilon$ small. Theorem 1.3 predicts that $d\tilde C_{p,2,K}=f_\varepsilon\,dH^1$ has exactly one convex solution, with support function $1-\varepsilon(2-p)^{-1}Y_{1,0}+O(\varepsilon^2)$; a second solution branch, a bifurcation at $\varepsilon=0$, or a sequence of bodies with nearly uniform $\tilde C_{p,2,K}$ whose diameters tend to infinity would refute the theorem.

Watch

Extended reading notes

Core claim

The central claim is Theorem 1.3: for $n\ge 2$, $\alpha\in(0,1)$ and $p\in(-1,1)$, there is an $\varepsilon_0>0$ such that whenever $|q-n|<\varepsilon_0$ and $\|f-1\|_{C^\alpha}<\varepsilon_0$, the measure equation $d\tilde C_{p,q,K}=f\,dH^{n-1}$ has a unique weak-measure solution $K\in K_o^n$, and $u=h_K|_{S^{n-1}}$ is a positive $C^{2,\alpha}$ solution of the corresponding Monge–Ampère equation. The proof fuses three ingredients: an optimal $C^0$ estimate (Theorem 1.6) bounding the diameter and volume of any body whose $L_p$ dual curvature is close to the spherical measure; a compactness argument showing such bodies converge to the unit ball, using the known uniqueness of the exactly isotropic solution; and local invertibility of the map $(q,u)\mapsto(q,(\|\nabla u\|^2+u^2)^{(q-n)/2}u^{1-p}\det(\nabla^2 u+uI))$ at $(n,1)$, whose linearization is $\Delta\varphi+(n-p)\varphi$ and is invertible precisely because $n-p>0$. In the even category the same scheme yields Theorem 1.5: uniqueness for $-1<p<q<\min\{n,n+p\}$ with $q>0$.

Load-bearing premise

The weakest link is the proof that any candidate body must stay bounded and cannot collapse to zero volume: that step is a contradiction argument whose punchline is a previously established theorem saying a certain flat limiting configuration cannot exist, and if that theorem does not cover the limit produced by the rescaling, the $C^0$ estimate and the uniqueness conclusion are unsupported.

Editorial extensions

If this is right

  • When $q=n$ and $p\in(-1,0)$, the theorem settles the near-isotropic $L_p$ Minkowski problem, the last open range after $p\in[0,1)$ had been handled previously.
  • For every $p\in(-1,1)$ and $q$ close enough to $n$, the problem is well-posed near $f\equiv 1$: the solution is unique, its support function is positive and $C^{2,\alpha}$, and it is $C^{2,\alpha}$-close to the constant function $1$ when $f$ is $C^\alpha$-close to $1$.
  • The range $-1<p<1$ is optimal for the $C^0$ estimate that carries the proof: for $p\le -1$ with $q=n$, examples exist of bodies with arbitrarily small volume whose $L_p$ surface-area density lies between two fixed positive constants.
  • In the even (origin-symmetric) setting, uniqueness holds for the larger window $-1<p<q<\min\{n,n+p\}$, $q>0$, so the even theory is not confined to the near-isotropic $q\approx n$ regime.
  • Because the proof identifies any limiting body as the unit ball, the uniqueness statement is accompanied by a continuity statement: as $f$ tends to $1$ in $C^\alpha$, the unique solution $K$ tends to the unit ball in the Hausdorff metric.

Reading between the lines

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

  • The same local-to-global scheme suggests a general recipe: for any family of curvature measures whose linearization at the constant solution is invertible and for which a $C^0$ estimate rules out escaping to infinity or degenerating to a lower-dimensional body, near-isotropic uniqueness should follow by the same argument.
  • The proof's blow-up analysis ties near-isotropic uniqueness to a purely structural statement — no constant $L_p$ surface-area density can be supported on a proper subspace — so any counterexample for $p\in(-1,1)$ would have to produce a degenerate limit whose density is not constant, bypassing that structural obstruction.
  • The optimality of the barrier $p=-1$ suggests a testable prediction: no norm on the perturbation $f-1$ with the same scaling behavior can restore near-isotropic uniqueness for $p\le -1$; a different mechanism would be needed there.
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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

2 major / 4 minor

Summary. The paper studies uniqueness in the Lp qth dual Minkowski problem near the isotropic case. The main result, Theorem 1.3, states that for n ≥ 2, α ∈ (0,1), and p ∈ (-1,1), there exists ε0 > 0 such that if |q - n| < ε0 and f ∈ C^α(S^{n-1}) is ε0-close in C^α to the constant function 1, then the equation d~Cp,q,K = f dH^{n-1} has a unique solution K ∈ K_o^n in the Alexandrov sense, and h_K|S^{n-1} is a positive C^{2,α} solution of (4). The proof combines new C0 estimates (Theorems 1.6 and 1.7) with a local inverse function theorem and a convergence argument in Section 7. The paper also proves Theorem 1.5, a near-isotropic uniqueness result for even solutions when -1 < p < q < min{n, n+p} and q > 0.

Significance. If correct, Theorem 1.3 is a substantial advance: it settles near-isotropic uniqueness for the full range p ∈ (-1,1), including the previously open case p ∈ (-1,0), and it extends the result from Lp surface-area measure to Lp qth dual curvature measures with q close to n. The C0 estimate with the sharp threshold p > -1, complementing the counterexamples of Jian-Lu-Wang, is an important contribution in its own right. The paper is generally carefully written, with detailed derivations of the anisotropic blow-up estimates and explicit statements of the dependence of constants. The main reservation concerns a gap in the application of Theorem 4.5 in the proof of Theorem 1.6(a), as detailed below; the bounded-density variant for n = 3, 4 is not affected by that gap.

major comments (2)
  1. [Section 4, equation (72)] The statement 'This fact contradicts Theorem 4.5(ii)' is not justified for p ≠ 0. Equation (72) identifies the Lp surface-area measure S_{p,K∞} with βH^{k-1} on L ∩ S^{n-1}, whereas Theorem 4.5(ii) is a statement about the cone-volume measure V_{K∞}. The conversion via formula (50) with q = n gives dV_{K∞} = (1/n) h_{K∞}^p dS_{p,K∞}, so the induced cone-volume density on L ∩ S^{n-1} is (β/n) h_{K∞}^p. For p ∈ (-1,1) \ {0}, constancy of this density is equivalent to constancy of h_{K∞} on L ∩ S^{n-1}, and that constancy is not proved anywhere in the argument. Since this contradiction is exactly what rules out lim_m diam K_m = ∞ in Theorem 1.6(a), the C0 estimate for p ≠ 0 lacks support as written. The gap is repairable: one could either prove that h_{K∞}|_{L ∩ S^{n-1}} is constant, or replace the appeal to Theorem 4.5(ii) by the standard fact that the surface-area measure of a full-dimensional convex body cannot be supported on a proper subsphere. For the bounded-density case n = 3,4, the same conversion is harmless because only positivity/boundedness below of the cone-volume density is needed.
  2. [Section 7, proof of Theorem 1.3] The passage from L∞ convergence of h_{K_m} to 1 to C^{2,α} convergence is asserted with a reference to 'Caffarelli's regularity theory and Schauder estimates (similar to the arguments in [23, Proof of Lemma 4.1])' and no details are given. This step is load-bearing because it is used to contradict (119) and to conclude that the solution is positive and C^{2,α}. The manuscript should either provide the bootstrap argument or state and prove the precise compactness/regularity lemma that applies under the hypotheses |q_m - n| → 0, f_m → 1 in C^α, together with the a priori bounds from Theorem 1.6. Section 7 is introduced as a sketch, but Theorems 1.3 and 1.5 are the main theorems of the paper, so this is a substantive omission rather than a purely expository one.
minor comments (4)
  1. [Abstract] The word 'Holder' should be spelled 'Hölder'.
  2. [Introduction, page 2] 'in the letter case' appears to be a typo for 'in the latter case'.
  3. [Section 4, Claim 4.1] The proof of Claim 4.1 refers to equation '(56)' when verifying estimate (54); the cross-reference should be to the displayed estimate being proved, or the numbering should be corrected.
  4. [Section 7] The proof of Theorem 1.5 is described as 'similar to that of Theorem 1.3' and then concludes via [21]; since the parameter regime and limiting equation differ, one or two sentences explaining how Theorem 1.7 and [21] combine would improve clarity.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the proof is a self-contained perturbation-compactness argument; self-citations are to independent prior theorems.

full rationale

The central claim (Theorem 1.3) is proved by linearizing the Lp dual Minkowski equation at (q,u)=(n,1), applying the inverse function theorem to obtain a local uniqueness neighborhood, and then using the C0 estimate (Theorem 1.6) plus Blaschke selection to force any near-isotropic solution into that neighborhood. The limiting step reduces to the isotropic equation S_{p,K∞}=H^{n-1}, whose unique solution is the unit ball by Brendle-Choi-Daskalopoulos [15], an external theorem not proved in this paper and not equivalent to the near-isotropic statement. The C0 estimate in Section 4 is an independent compactness result; its proof uses an anisotropic blow-up and derives that the limiting Lp surface-area measure is a constant multiple of H^{k-1} on a subspace L. The contradiction is then attributed to Theorem 4.5, quoted from Saroglou [85] and Boroczky-Saroglou [14]. This is a self-citation and it is load-bearing, but it is a parameter-free non-existence theorem with stated hypotheses that do not include the present result, so it qualifies as independent support. A possible concern that converting (72) into a cone-volume density requires inserting an unproved factor h^p_{K∞} would be a mathematical gap, not a circular reduction: no equation in the paper is defined in terms of the theorem being proved, no fitted parameter is renamed as a prediction, and no known result is merely recast in new coordinates. The paper is not circular.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

No free parameters are fitted; the constants ε0, C, Γ are existential. No new entities are postulated. The proof relies on standard convex-geometric tools and several prior uniqueness / non-existence theorems, in particular Theorem 4.5 from the authors' own previous work.

assumptions (6)
  • standard math Caffarelli's regularity theory for Monge-Ampere equations (Theorem 3.1)
    Invoked in Lemma 3.2 to conclude h_K is C^1 on {h_K>0} and C^{2,α} when f is C^α; this underlies all measure-to-function conversions in the proof.
  • standard math Brendle-Choi-Daskalopoulos uniqueness of the isotropic Lp Minkowski solution for -n<p<1
    Used in Section 7 to identify the limit body in the compactness argument as the unit ball.
  • standard math Chen-Huang-Zhao uniqueness of even isotropic Lp qth dual Minkowski solutions for p≥-n, q≤min{n,n+p}, q≠p
    Used in Section 7 for Theorem 1.5 to identify the limit of even solutions.
  • standard math Theorem 4.5 (non-existence of constant cone-volume density on a subspace) from Saroglou and Boroczky-Saroglou
    Load-bearing in the proof of Theorem 1.6; the contradiction in the blow-up argument relies on it.
  • standard math John/KLS ellipsoid inclusion E⊂K⊂nE for every convex body
    Used throughout Sections 4-6 to parametrize bodies by ellipsoid axes.
  • standard math Spectral decomposition of the spherical Laplacian, eigenvalues k(k+n-2)
    Used in Section 7 to show L_u0 = Δ + (n-p) is invertible for -1<p<1.

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Cite this review

Pith. "Pith review of Uniqueness in the near isotropic Lp dual Minkowski problem." pith.science (2026). https://pith.science/paper/2OLRSBCO

@misc{pith2026250501066,
  author       = {Pith},
  title        = {Pith review of: Uniqueness in the near isotropic Lp dual Minkowski problem},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2OLRSBCO}},
  note         = {Machine review of arXiv:2505.01066}
}
read the original abstract

For n>1 and -1<p<1, we prove that if q is close to n and the qth Lp dual curvature is Holder close to be the constant one function, then this "near isotropic" qth Lp dual Minkowski problem on the (n-1)-dimensional sphere has a unique solution. Along the way, we establish a C0 estimate for -1<p<1 that is optimal in the sense that if p<-1 and q=n, then it is known that the analogous C0 estimate does not hold. We also prove the uniqueness of the solution of the near isotropic even qth Lp dual Minkowski problem on the (n-1)-dimensional sphere if -1<p<q<min{n,n+p} and q>0.

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Forward citations

Cited by 2 Pith papers

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

  1. The uniqueness and non-uniqueness of solutions to the even dual Minkowski problem

    math.AP 2026-08 accept novelty 8.0 of 10

    In the plane, for 0<q<2, equal dual curvature measures force origin-symmetric convex bodies to coincide; for q>n in R^n, distinct origin-symmetric bodies can share the same dual curvature measure.

  2. The Lp centro-sectional Minkowski problem

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    The paper formulates the Lp centro-sectional Minkowski problem, solves it for p>1 and q>0, establishes regularity and uniqueness, and proves Lp Brunn-Minkowski-type inequalities for large p.

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