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REVIEW 3 major objections 7 minor 94 references

Compactness of the $L_p$ dual Minkowski problem in $\mathbb{R}^3$

T0 review · 3 major / 7 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read Bounded positive $L_p$ dual curvature on the sphere forces every solution body in $\mathbb{R}^3$ to have controlled size.

desk verdict Theorem 1.2 is a genuine extension with a sound geometric strategy, but the proof rests on Lemma 3.2, which is asserted without proof; the paper deserves a referee who insists that lemma be fixed. read the letter →

arxiv 2505.17219 v1 pith:GUS73M4B submitted 2025-05-22 math.AP

Karoly J. Boroczky , Shibing Chen , Weiru Liu , Christos Saroglou This is my paper · ORCID
classification math.AP MSC 35J9652A20
keywords L_pdualMinkowskiproblemcurvaturemeasureMonge-AmpèreequationC^0estimateconvexbodyJohnellipsoiduniquenesscompactness
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 establishes a $C^0$ a priori estimate for the $L_p$ $q$th dual Minkowski problem on the two-dimensional sphere. Theorem 1.2 states that for $p\in[0,1)$ and $q>2+p$, any convex body $K\subset\mathbb{R}^3$ containing the origin whose $L_p$ $q$th dual curvature measure lies between $\lambda^{-1}H^2$ and $\lambda H^2$ must satisfy $\sup h_K\le C$ and $|K|\ge C^{-1}$, with $C$ depending only on $\lambda,p,q$. This converts a bound on a prescribed measure into geometric compactness of the solution family, the standard prerequisite for existence and uniqueness arguments in Monge\,Amp\`ere theory. As a corollary, the paper proves uniqueness of the near-isotropic solution when $q$ is close to $3$ and the prescribed density is H\"older-close to the constant function $1$. The parameter restriction is meaningful: the paper records that for $p<-1$ and $q=3$ the analogous estimate fails.

What carries the argument

The carrying mechanism is Lemma 3.2, a regularity and measure-conversion result that turns an Alexandrov-sense solution $d\tilde C_{p,q,K}=f\,dH^2$ into the pointwise identities $$d\tilde C_{p,q,K}=$3h_K^{{-p}}$\|Dh_K\|^{q-3}\,dV_K=$h_K^{{1-p}}$\|Dh_K\|^{q-3}\,dS_K,\qquad dV_K=\tfrac13 h_K^p\|Dh_K\|^{3-q}\,d\tilde C_{p,q,K}.$$ These identities let every later estimate trade prescribed dual curvature against cone-volume measure and surface-area measure on selected regions of the sphere. Around them the proof uses the John ellipsoid (the maximal-volume ellipsoid inscribed in $K$), whose containment $E\subset K\subset X+3(E-X)$ and half-axes $r_1\le r_2\le r_3$ organize the case analysis, and derives the basic estimate $r_1r_2r_3\approx r_3^{3-q+p}$ by integrating the identities over a cap near the longest axis. Section 5 then consists of upper and lower bounds on $\tilde C_{p,q,K}(F_\pm)$ for caps $F_\pm$ defined by the suspected degeneration, each pair contradicting the fixed density $f$.

What would settle it

Check whether the counterexample family cited for $p<-1$, $q=3$ can be adapted to some $p\in[0,1)$, $q>2+p$; if such a sequence of bodies with bounded density and vanishing volume exists, Theorem 1.2 is false. More locally, test Lemma 3.2: an Alexandrov solution of $d\tilde C_{p,q,K}=f\,dH^2$ with a flat facet containing the origin whose outward normals form a positive-area subset of $S^2$ would contradict the lemma's zero-area claim and invalidate the identities (41)--(43).

Watch

Extended reading notes

Core claim

On its own terms, the paper's discovery is a compactness theorem for the $L_p$ dual Minkowski problem in $\mathbb{R}^3$: in the range $p\in[0,1)$, $q>2+p$, a uniform two-sided bound on the dual curvature measure forces a uniform bound on the body from both above and below. The proof is by contradiction. Given a sequence of bodies that should violate the theorem, the proof places their John ellipsoids, orders the half-axes $r_1\le r_2\le r_3$, and uses the measure identities of Lemma 3.2 to read the bounded-density condition as a relation among the support function, the radial length of its gradient, and the cone-volume measure. The basic estimate $r_1r_2r_3\approx r_3^{3-q+p}$ follows, implying $r_3\gtrsim 1$ and $r_1\lesssim 1$; the remaining work rules out the two possible degenerations $r_1\le r_2\ll r_3$ and $r_1\ll r_2\approx r_3$ by finding a spherical cap whose dual curvature mass is forced to be simultaneously large and small.

Load-bearing premise

The proof depends on Lemma 3.2, stated but not proved in the paper, which asserts that a solution body is regular on the set where its support function is positive and that the dual curvature measure can be written in the pointwise identities (41)--(43), including the claim that the directions in which the origin is a boundary normal form a set of zero area; if this unproved bridge fails, every estimate in Sections 4 and 5 loses its starting point.

Editorial extensions

If this is right

  • For $p\in[0,1)$, $q>2+p$, any family of solution bodies with densities between $\lambda^{-1}$ and $\lambda$ has uniformly bounded diameter and uniformly bounded volume from below, so a subsequence converges to a convex body in $\mathbb{R}^3$ containing the origin.
  • Corollary 1.3 follows: for $q$ sufficiently close to $3$ and $f$ H\"older-close to $1$, the equation $d\tilde C_{p,q,K}=f\,dH^2$ has exactly one solution $K\in\mathcal{K}_o^3$, and its support function is a positive $C^{2,\alpha}$ function on $S^2$.
  • The condition $q>2+p$ enters through the positive exponent $q-p-2$ that dominates the measure estimates on caps; relaxing it would break the contradiction arguments in Lemmas 5.2, 5.3, and 5.7.
  • The known counterexamples for $p<-1$, $q=3$ show the theorem's lower restriction on $p$ is necessary; the estimate cannot hold across the full range of $p$ in Problem 1.1.

Reading between the lines

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

  • Editorial inference: the $C^0$ estimate should make existence of solutions for arbitrary positive bounded densities in this parameter range approachable by standard continuity or Gauss-curvature-flow arguments, a step the paper itself does not take.
  • Editorial inference: the two-case degeneration analysis is specific to three dimensions; moving to $S^{n-1}$ with $n\ge 4$ would presumably require a more complex classification of thin bodies, which may be why the theorem is stated only for $\mathbb{R}^3$.
  • Editorial inference: the proof is by contradiction and yields no explicit constants or rates; a quantitative version of Theorem 1.2 would give stability of the near-isotropic solution, not merely its uniqueness.
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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 / 7 minor

Summary. The paper proves a C^0 compactness estimate for the L_p q-th dual Minkowski problem on S^2. Theorem 1.2 states that for p in [0,1), q > 2+p and λ > 1, if a convex body K in R^3 containing the origin satisfies λ^{-1}H^2 ≤ eC_{p,q,K} ≤ λ H^2, then sup h_K ≤ C and |K| ≥ C^{-1}. The proof uses the John ellipsoid, a basic volume-scale estimate (Lemma 4.1), and a contradiction argument eliminating degenerate limiting shapes in two cases: a flat pancake (Case I) and a thin cigar (Case II). A uniqueness corollary near the isotropic case is also stated. The central claim is Theorem 1.2, and all later sections depend on the measure-density identities in Lemma 3.2.

Significance. If the proof is completed, Theorem 1.2 would resolve Problem 1.1 in the three-dimensional case for p in [0,1) and q > 2+p, providing the first compactness estimate in this parameter range without symmetry assumptions. The geometric case analysis with explicit cap estimates and volume comparisons is a substantial contribution, and the argument contains no fitted parameters. However, the proof is conditional on Lemma 3.2, which is asserted without proof; since this lemma supplies the pointwise identities used in every estimate of Sections 4 and 5, the result is not yet fully established.

major comments (3)
  1. [Section 3, Lemma 3.2] Lemma 3.2 is the single bridge from the Alexandrov measure equation to the density identities (41)-(43) and to the assertion (iv) that H^2(ν_K(O)) = 0 when O ∈ ∂K. All estimates in Sections 4 and 5 use these identities, for example (45) in Section 4 and (42) in Lemmas 5.1-5.7. The text states that the lemma follows from Caffarelli's theorem and an observation, but no proof is given. In particular, the verification that the right-hand side of the local equation (38) satisfies the hypotheses of Theorem 3.1 is not routine: for q > 3 the factor (∥Dv∥^2 + (⟨Dv,y⟩-v)^2)^{(3-q)/2} is singular unless the radial function is already known to be bounded below away from zero, and for p < 1 the factor v^{p-1} is unbounded as h_K approaches 0. Moreover, part (iv) is not a local consequence of (38), which is set up only at points where h_K(e) > 0, and the remark after Lemma 3.2 (citing [8, Example 4.2]) shows that Γ_K can contain a circular disk centered at O, so the borderline assertion is exactly what needs a proof. A complete proof of Lemma 3.2, or a precise reference with all hypotheses verified, is required before Theorem 1.2 can be considered established.
  2. [Section 5.1, Lemma 5.1] The lower bound V_K(F_1) ≥ C_2 r_1 r_2 r_3 in (65) is justified by the sentence 'the argument leading to [25, estimate (3.7)] shows ...' without stating the result or verifying its hypotheses in the present setting. Since (65) is used to rule out Subcase (i) of Case I, and since [25] is co-authored by two of the current authors (Chen and Liu), this imported estimate is load-bearing. The authors should either prove the estimate directly or state the exact lemma from [25] and show that the geometric conditions of that lemma hold here, in particular that the constant C_2 is independent of M under the assumption (61).
  3. [Section 5.2, Lemma 5.7] The derivation of the key width estimate (123)-(124) is compressed. After (125), the bound r_3^{2-3(q-p)} ≲ r_3^{2-q+p}(r_1/r_3)^2 is used, which appears to require the intermediate inequality r_3^{-(q-p)} ≲ r_1/r_3 from Corollary 4.2 together with q > p+2 and r_3 ≳ 1; the algebra is not shown and the reader must check the exponents. In addition, the passage from (123) or (124) to the dichotomy (126) or (127) uses a sign condition, either ν_2^{(m)}/ν_1^{(m)} + ξ_2^{(m)}/|ξ_1^{(m)}| ≥ 0 or ≤ 0, but the proof does not state that the sign is eventually constant along a subsequence. This is fixable by a subsequence argument, but as written the contradiction at the end of Lemma 5.7 relies on a choice that is not fully justified. Please expand this part so that the sign choice is uniform in m.
minor comments (7)
  1. [Abstract] The word 'curvarture' in the abstract is a typo and should be 'curvature'.
  2. [Section 3] Theorem 3.1 is introduced as 'Caffarelli' but then referred to as 'Caffarelli's Lemma 3.1' in the text before Lemma 3.2; please unify the numbering and the theorem/lemma terminology.
  3. [Corollary 1.3] The text states that Corollary 1.3 was already proved in [10] by the same authors, yet the paper presents it as a corollary of Theorem 1.2. Please clarify whether this is a new result or a previously proved theorem included for context, and adjust the introduction accordingly.
  4. [Section 2, equation (38)] In the displayed local equation (38), the notation ∥Dv∥^2 + (⟨Dv,y⟩ - v)^2 should be checked against the definition of ∥Dh_K∥ in Section 2; the sign conventions and the exponent in the factor (1+∥y∥^2)^{-(3+p)/2} should be verified explicitly.
  5. [Figures] Figures 2 and 3 are referenced in the proofs of Lemmas 5.1 and 5.2, but in the posted version the figures appear to be missing or not labeled in the text; please ensure that all figures are included and referenced.
  6. [Lemma 5.5, statement] The statement 'there exists a unit vector ν ∈ ν_K(G_+)' is imprecise because ν_K is a set-valued map; it should read 'there exists x ∈ G_+ and a unit vector ν ∈ ν_K(x)'.
  7. [References] Reference [10] is cited as an unpublished manuscript; since it is used for Corollary 1.3 and for prior cases of Problem 1.1, please provide an arXiv identifier or state 'in preparation'.

Circularity Check

2 steps flagged · score 2.0 of 10

Minor self-citations in auxiliary claims, but the central C0 estimate is derived self-contained; no circular reduction.

  1. self citation load bearing [Section 5.1, Lemma 5.1 proof, near equation (65)]
    "On the other hand, the argument leading to [25, estimate (3.7)] shows that S{conv{o, x} : x ∈ G1} contains a translated copy of ~c0E where ~c0 ∈ (0, 1) depends on c0, λ, p, q, and hence (cf. (11)) VK(F1) = |[ {conv{o, x} : x ∈ G1} | ≥ C2r1r2r3"

    Reference [25] (Chen-Feng-Liu) shares two co-authors with the present paper (S. Chen and W. Liu). The lower bound VK(F1) ≥ C2 r1 r2 r3 is not proved in the text; it is imported via 'the argument leading to [25, estimate (3.7)]'. This bound is load-bearing for excluding Subcase (i) of Case I: the subsequent contradiction (67) depends on it. However, the imported content is a cap-volume comparison from a distinct published paper, not an equivalent reformulation of Theorem 1.2, so this is a self-citation carrying one branch of the proof rather than a circular reduction of the main theorem.

  2. self citation load bearing [Section 1, after Corollary 1.3]
    "Using the method in Böröczky, Chen, Liu, Saroglou [10], one obtains the following uniqueness result based on Theorem 1.2. ... Actually, Böröczky, Chen, Liu, Saroglou [10] prove Corollary 1.3, and its analogues for any n ≥ 3."

    The paper advertises Corollary 1.3 as a consequence obtained 'based on Theorem 1.2', but the very next sentence attributes the proof to the companion paper [10] by the same four authors. Thus the uniqueness conclusion is not derived in this paper; it is delegated to a self-citation. This does not bear on the compactness estimate in Theorem 1.2, but it is a direct instance of a stated result resting on the authors' own companion work.

full rationale

The main compactness estimate (Theorem 1.2) is not circular. The proof starts from the measure equation d eCp,q,K = f dH2 and proceeds by: (i) invoking Caffarelli's classical Monge-Ampere regularity through Lemma 3.2 to obtain the density identities (41)-(43); (ii) introducing the John ellipsoid and proving the basic estimate r1r2r3 ≈ r3^{3-q+p} (Lemma 4.1) via volume and curvature comparisons; and (iii) excluding the two possible degenerate axis-length configurations in Section 5 by geometric normal-set estimates together with the lower bound f ≥ 1/λ. No parameter is fitted to the quantity being bounded, and no conclusion of Theorem 1.2 is used to define or justify its own hypotheses. The two self-citations found are auxiliary: Lemma 5.1 imports an argument/estimate from the co-authored paper [25] to control cone volume in one subcase, and Corollary 1.3 is explicitly attributed to the authors' companion paper [10]. Neither citation is equivalent to Theorem 1.2, and the C0 estimate has independent mathematical content. One separate rigor concern, which is not circularity: Lemma 3.2 is stated without proof, and part (iv) (H2(νK(O)) = 0 when O ∈ ∂K) is not visibly a consequence of the displayed local equation (38); this is an unverified bridge and a correctness risk, but not a case of the derivation reducing to its own inputs.

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

No free parameters and no invented entities are needed. The proof uses standard results in convex geometry and Monge-Ampere theory, plus one published self-cited geometric estimate in Lemma 5.1. Corollary 1.3 is already proved in the authors' companion paper [10], but that result is not used to prove Theorem 1.2.

assumptions (5)
  • standard math Caffarelli regularity: if det(D^2 v)=f with 1/lambda <= f <= lambda on a convex domain, then v is strictly convex and locally C^{1,alpha}; with C^{0,beta} f, v is locally C^{2,beta}.
    Theorem 3.1 imports this result and it is the basis for Lemma 3.2, which all later estimates rely on.
  • standard math John ellipsoid theorem: the maximal volume ellipsoid E in K satisfies E subset K subset X+3(E-X), and the half axes are ordered r1 <= r2 <= r3.
    Section 4 starts from this theorem to define the axes r_i and to prove the basic estimate r1r2r3 approximately r3^{3-q+p} in Lemma 4.1.
  • standard math Blaschke selection theorem: any bounded sequence of convex bodies has a subsequence converging in Hausdorff distance.
    Used in Lemma 5.7 to obtain the planar limit body K_infty after blowing down the rescaled bodies.
  • standard math Support function and radial function calculus, including identities (5)-(8), (21)-(27), and (41)-(43).
    These identities connect support functions, radial functions, surface area measures, and cone volume measures; they are used throughout Sections 4 and 5.
  • domain assumption The geometric estimate [25, estimate (3.7)]: the set conv{o,x}: x in G1 contains a translated copy of c0 E for a constant c0 independent of M.
    Lemma 5.1 imports this estimate from a paper by Chen-Feng-Liu, which overlaps with the current authors. It is published and is not equivalent to Theorem 1.2, but it is a nontrivial self-cited ingredient.

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Cite this review

Pith. "Pith review of Compactness of the $L_p$ dual Minkowski problem in $\mathbb{R}^3$." pith.science (2026). https://pith.science/paper/GUS73M4B

@misc{pith2026250517219,
  author       = {Pith},
  title        = {Pith review of: Compactness of the $L_p$ dual Minkowski problem in $\mathbbR^3$},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/GUS73M4B}},
  note         = {Machine review of arXiv:2505.17219}
}
abstract

We prove the $C^0$ estimate for the $L_p$ $q$th dual Minkowski problem on $S^2$ under fairly general conditions; namely, when $p$ lies in [0,1) and $q>2+p$, and the $L_p$ $q$th dual curvarture is bounded and bounded away from zero. We note that it is known that the analogous $C^0$ estimate does not hold if $p<-1$ and $q=3$. As a corollary of our $C^0$ estimate, we deduce the uniqueness of the solution of the near isotropic $q$th $L_p$ dual Minkowski problem on $S^2$ if $q$ is close to 3 and the $q$th $L_p$ dual curvature is Holder close to be the constant one function.

Figures

Figures reproduced from arXiv: 2505.17219 by the authors.

Figure 1
Figure 1. lemma 4.1 y = (y1, y2, y3) ∈ ∂K\G and νy = (νy,1, νy,2, νy,3) ∈ νK(y), by convexity we have ⟨Y −y, νy⟩ ≤ 0. By the choice of Y and y, we have Y3−y3 > 3 4 r3 > 0. It follows from (47) that y1 − Y1 ≤ 12r1 and y2 − Y2 ≤ 12r2. Hence, if νy,1 > 0 and νy,2 > 0, then νy,3 ≤ y1 − Y1 Y3 − y3 · νy,1 + y2 − Y2 Y3 − y3 · νy,2 ≤ 16 · νy,1 + 16 · νy,2. Let S100 := {ν = (ν1, ν2, ν3) ∈ S 2 : ν3 ≤ 100 · ν1 + 100 · ν2}, and S 2 + := … view at source ↗
Figure 2
Figure 2. lemma 5.1 Now Case I can be subdivided into the following two subcases: Subcase (i): for some constant c0 ∈ (0, 1 3 ), for any M > 0, there exists a convex body K ∈ K3 o satisfying (44), (59) and dist(O, ∂I) ≥ c0r3. (61) Subcase (ii): for any M > 1, there exists a convex body K ∈ K3 o satisfying (44), (59) and dist(O, ∂I) < r3 M . (62) LEMMA 5.1. Subcase (i) in Case I does not occur. Proof. In this argument, the imp… view at source ↗
Figure 3
Figure 3. lemma 5.2 We deduce that ν3 ≤ 96r2 δr3 = 1 4 by δ = 384r2 r3 , which in turn implies F1 ⊂ S 1 4 . It follows that G := n x ∈ ∂K : νK(x) ∈ S 2 \S 1 4 o ⊂ ∂K\G1. For H2 a.e. ν ∈ S 2\S 1 4 , we have that DhK(ν) ∈ ∂K\G1. Thus, (62), (59) and the definiton of G1, yield that ||DhK(ν)|| ≲ r3 M . (70) Since δ = 384r2 r3 , using the assumption r1 ≈ r2 ≪ r3 (cf. (68)), we have H2 (G) ≤ H2 (∂K\G1) ≲ δr2r3 + r1r2 ≲ r 2 1 . (71)… view at source ↗
Figures from the paper (3 more)
Figure 4
Figure 4. Figure 4: lemma 5.3 where δ = 384r2 r3 converges to zero as M converges to +∞. On the one hand, similar to the proof of (69), for any ν = (ν1, ν2, ν3) ∈ F1, we have ν3 ≤ 96r2 δr3 = 1 4 , which implies F1 ⊂ S 1 4 :=  ν ∈ S 2 : ⟨ν, e3⟩ ≤ 1 4  . Denote by K′ the projection of K o…
Figure 5
Figure 5. Figure 5: lemma 5.4 Proof. In the proof of Lemma 5.4, the implied constants in ≲ and ≈ depend also on the C1 and c4 introduced in (81) and (82) besides λ, p, q. For any object Y ⊂ R 3 , we write Y ′ to denote its projection in the coordinate plane lin{e2, e3}. We suppose that su…
Figure 6
Figure 6. Figure 6: lemma 5.7 It follows from fm ≥ 1 λ and the definition of Fε,+ and Fε,− that the density function fm of Cep,q,Km satisfies Z Fε,+ fm dH2 ≥c0θε,+ (116) Z Fε,− fmdH2 ≥c0θε,− (117) where c0 ∈ (0, 1) depends on λ, and is independent of ε and m. Our next goal is to find some…

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