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

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Pith's one-line read This paper proves that the even dual Minkowski problem has a unique origin-symmetric planar solution for 0<q<2, but admits two different smooth solutions for every q>n≥2.

desk verdict Settles planar even dual Minkowski uniqueness for 0<q<2 and extends non-uniqueness to q>n; the proofs are careful and the external equality input checks out. read the letter →

arxiv 2608.01912 v1 pith:3P54GXY5 submitted 2026-08-03 math.AP math.DG

Xiong Ge , Yang Kaiwen This is my paper · ORCID
classification math.APmath.DG MSC 52A3835J96
keywords convexbodydualMinkowskiproblemcurvaturemeasurequermassintegrallogarithmicBrunn-MinkowskiinequalityMonge-Ampèreequationuniquenessnon-uniqueness
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 settles two ends of the uniqueness question for the even dual Minkowski problem, which asks whether a convex body is determined by its dual curvature measures. The authors prove that in the plane, for exponents $0n$ in every dimension $n\ge 2$, uniqueness fails: different origin-symmetric bodies can carry the same dual curvature measure, and the two bodies can be chosen smooth, so the associated spherical Monge–Ampère equation has at least two classical solutions. Both results are governed by a logarithmic Brunn–Minkowski inequality for dual quermassintegrals that the paper establishes in the plane and shows to fail for $q>n$.

What carries the argument

The central object is the $q$-th dual curvature measure $\tilde C_q(K,\cdot)$ and its total mass, the $q$-th dual quermassintegral $\tilde V_q(K)=\frac1n\int_{S^{n-1}}\rho_K(u)^q\,du$, where $\rho_K$ is the radial function. The mechanism is the logarithmic Brunn–Minkowski inequality for these integrals, $\tilde V_q((1-\lambda)\cdot K+_0\lambda\cdot L)\ge \tilde V_q(K)^{1-\lambda}\tilde V_q(L)^\lambda$, with the Wulff shape $(1-\lambda)\cdot K+_0\lambda\cdot L=[h_K^{1-\lambda}h_L^\lambda]$ as the interpolation. The paper derives the planar inequality for $0<q<2$ by a one-dimensional Prékopa–Leindler step applied to the layer-cake formula $\tilde V_q(A)=\frac{q(n-q)}{n}\int_0^\infty r^{q-n-1}V_n(A\cap rB^n)\,dr$, and extracts equality from the equality cases of the volume logarithmic Brunn–Minkowski inequality. For the non-uniqueness range $q>n$, the same Wulff-shape machinery is used in reverse: an explicit family of rectangles $A_t=[-(1+t),1+t]\times[-(1-t),1-t]$ satisfies $\tilde V_q(A_t)=\tilde V_q(A_{-t})$ while $\tilde V_q([(h_{A_t}h_{A_{-t}})^{1/2}])$ is strictly smaller than the geometric mean, and taking Cartesian products with small cubes lifts the strict failure to $\mathbb{R}^n$.

What would settle it

At $q=1$, solve the even dual Minkowski problem for an arbitrary smooth positive even density $f$: find two distinct origin-symmetric smooth convex bodies $K,L$ satisfying $\tilde C_1(K,\cdot)=\tilde C_1(L,\cdot)=f\,du$. The paper's Theorem 1.1 says no such pair exists; producing one would refute the uniqueness claim.

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Extended reading notes

Core claim

The paper's central discovery is a sharp transition in the exponent $q$ for the even dual Minkowski problem. For $n=2$ and $0<q<2$, the problem is well-posed among origin-symmetric convex bodies: $\tilde C_q(K,\cdot)=\tilde C_q(L,\cdot)$ implies $K=L$. For $q>n\ge 2$, it is not: there exist origin-symmetric convex bodies $K,L$, smooth and uniformly convex, with $\tilde C_q(K,\cdot)=\tilde C_q(L,\cdot)$ but $K\neq L$. Equivalently, the spherical Monge–Ampère equation $h(u)|\nabla h(u)|^{q-n}\det(h_{ij}(u)+h(u)\delta_{ij})=f$ admits two different classical solutions for a suitable smooth positive even density $f$. The proof is driven by the logarithmic Brunn–Minkowski inequality for the dual quermassintegrals $\tilde V_q$: the paper proves it in $\mathbb{R}^2$ for $0<q<2$ with equality only for dilates, and disproves it for $q>n$ using an explicit family of rectangles, lifted to higher dimensions.

Load-bearing premise

The planar uniqueness result rests on an external theorem that characterizes exactly when equality holds in the logarithmic Brunn–Minkowski inequality; if that characterization has hidden exceptional cases, the conclusion that equal dual curvature measures force the bodies to be identical could fail.

Editorial extensions

If this is right

  • For $n=2$ and $0<q<2$, the even dual Minkowski problem is unique among origin-symmetric bodies: the data determine the body.
  • For every $q>n\ge 2$, the even dual Minkowski problem admits at least two distinct smooth origin-symmetric solutions with the same data, so well-posedness fails in this range.
  • The spherical Monge–Ampère equation $h(u)|\nabla h(u)|^{q-n}\det(h_{ij}(u)+h(u)\delta_{ij})=f$ has a smooth positive even density $f$ with at least two classical solutions for each $q>n$.
  • The explicit rectangle construction shows the logarithmic Brunn–Minkowski inequality for dual quermassintegrals is genuinely false for $q>n$, not merely unproved, so any uniqueness argument in this range must bypass that inequality.

Reading between the lines

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

  • If the volume logarithmic Brunn–Minkowski inequality with its conjectured equality cases is eventually proved in higher dimensions, the truncation argument in Lemma 3.3 would extend the planar uniqueness conclusion for $0<q<n$ to those dimensions; the present paper only establishes the planar case.
  • The boundary exponent $q=n$ is natural: for $n=2$ and $q=2$ the equality theory already gives uniqueness, and the local expansion in Section 4 changes sign exactly when $q$ passes $n$, leaving $q=n$ as the delicate threshold the paper does not address.
  • The non-uniqueness is probably not confined to the one-parameter rectangle family; any pair of origin-symmetric bodies whose Wulff combination has dual quermassintegral smaller than the geometric mean and whose radial integrals are matched by the $t\mapsto -t$ symmetry would produce another pair, so examples may be abundant.
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Editorial analysis

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

Referee Report

0 major / 6 minor

Summary. The paper studies uniqueness and non-uniqueness for the even dual Minkowski problem associated with the dual curvature measures \tilde C_q. The main results are: in \mathbb R^2, for 0<q<2, origin-symmetric convex bodies with equal dual curvature measures must coincide (Theorem 1.1); and for q>n\ge 2, there exist distinct origin-symmetric bodies with the same dual curvature measure (Theorem 1.2), with the construction upgraded to smooth bodies in Remark 4.2. These results are obtained by establishing a logarithmic Brunn–Minkowski inequality for dual quermassintegrals in the plane with equality characterization (Theorem 1.3), and by constructing explicit parallelotope counterexamples showing that the corresponding inequality fails for q>n (Theorem 1.4). The proof transfers the planar log-Brunn–Minkowski inequality to dual quermassintegrals through a Prékopa–Leindler layer-cake argument, uses concavity and variational arguments for the uniqueness part, and combines explicit rectangle counterexamples with a compactness argument for the non-uniqueness part.

Significance. If the results are correct, the paper substantially sharpens the known non-uniqueness range for the even dual Minkowski problem from q>2n to the borderline q>n, and it provides the first global uniqueness statement for positive q in the plane, for 0<q<2. The arguments are well-structured and mostly self-contained: the layer-cake transfer in Lemma 3.3 is elegant, the equality analysis uses a multi-scale truncation argument that appears to rule out the parallelogram exceptional cases of the classical planar log-Brunn–Minkowski inequality, and the counterexample in Theorem 1.4 is explicit and computable. The paper also gives a concrete failure of the conjectured dual log-Brunn–Minkowski inequality, which is a falsifiable statement in its own right. The exposition is clear, and the main computations in Sections 3 and 4 are reproducible.

minor comments (6)
  1. [Section 3, Lemma 3.3 and Theorem 1.3] The proof of Theorem 1.3 relies on the equality characterization of the planar log-Brunn–Minkowski inequality from [5], but Lemma 3.3 is stated conditionally on the direct-sum equality condition and the proof of Theorem 1.3 cites [5] only for the inequality. Please state explicitly the theorem from [5] that supplies the equality cases used here, or add a short proof of the needed equality cases for the truncated bodies K\cap rB^n and L\cap e^\tau rB^n; this is a load-bearing input for Theorem 1.1.
  2. [Section 2, Lemma 2.2] The notation lin(K_i-K_i) is used without definition; please explain that it denotes the linear subspace parallel to the affine hull of the set K_i-K_i.
  3. [Section 3, Lemma 3.3, Step 2] The phrase 'Lets s→∞ in the volume identity' contains a typo and should read 'Letting s→∞ in the volume identity'.
  4. [Section 4, Theorem 1.4, Step 2] The sentence 'Since the equality holds at all coordinate directions ±e_i, i=1,...,n, every point of [(h_{K_{t,\varepsilon}}h_{K_{-t,\varepsilon}})^{1/2}] lies in S_{t,\varepsilon}' is terse; it would be clearer to note explicitly that the defining inequalities at v=±e_i give |x_i|\le r_t for i=1,2 and |x_i|\le \varepsilon for i=3,\ldots,n.
  5. [Abstract and Introduction] The phrase 'our established logarithmic Brunn-Minkowski inequality for dual quermassintegrals' should be rephrased, for example as 'the logarithmic Brunn-Minkowski inequality for dual quermassintegrals established in this paper,' to avoid ambiguity about what is new.
  6. [Remark 4.2] Please change 'has different classical solutions' to 'has two different classical solutions' to match the statement that two distinct bodies are produced.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the derivation chain is self-contained and the load-bearing inputs are external theorems, not fitted parameters or self-citations.

full rationale

The paper's claimed results are obtained by legitimate mathematical arguments rather than by construction or by renaming inputs. Theorem 1.3 derives the dual log-Brunn–Minkowski inequality for 0<q<2 from the volume log-Brunn–Minkowski inequality of Böröczky–Lutwak–Yang–Zhang via a layer-cake representation (Lemma 3.1) and a Prékopa–Leindler argument (Lemma 3.3), with equality handled using the external direct-sum equality characterization. Theorem 1.1 then follows by a standard concavity/variational argument, not by assuming the conclusion. The non-uniqueness results are built on an explicit family of rectangles and parallelotopes (Theorem 1.4) with a direct computation of strict violation of the dual log-Brunn–Minkowski inequality, and Theorem 1.2 is a valid contradiction argument: assuming a hypothetical uniqueness of the Φ-functional would force the log-Brunn–Minkowski inequality, contradicting the explicit example. No parameter is fitted, no quantity is defined in terms of the target result, and the paper contains no self-citations by the present authors. The only load-bearing external input is the planar log-Brunn–Minkowski inequality and its equality characterization from [5]; this is an independent published theorem, so its use is evidence rather than circularity. Thus the paper is not circular.

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

The central claims are pure mathematics. The paper relies on published theorems from convex geometry: the planar log-BM inequality with equality cases ([5]), the direct-sum characterization ([4, Lemma 1]), the Prékopa-Leindler inequality, and the variational machinery of the dual Minkowski problem ([21]). No ad hoc assumptions are made; the only structural condition, q>n in Lemma 4.1, is part of the theorem's hypothesis.

assumptions (5)
  • standard math Planar logarithmic Brunn-Minkowski inequality with equality cases (Böröczky-Lutwak-Yang-Zhang [5])
    Invoked in Lemma 3.3 to transfer the volume log-BM inequality to dual quermassintegrals \tilde V_q, and in the equality analysis; the truncation argument needs the full equality characterization from [5].
  • standard math Direct-sum decomposition characterization via spherical image (Böröczky-Kalantzopoulos [4, Lemma 1])
    Used in Lemma 3.2 and the equality discussion to conclude that truncations K∩rB^n are not direct sums and that equality forces dilates.
  • standard math One-dimensional Prékopa-Leindler inequality with equality condition (Lemma 2.1, see [15])
    Core averaging tool in Lemma 3.3 connecting the volume log-BM inequality for truncated bodies to the dual quermassintegral inequality.
  • standard math Variational machinery of the dual Minkowski problem (Huang-Lutwak-Yang-Zhang [21], Lemmas 2.3 and 2.5)
    Used in Theorem 1.1 Step 1 to derive the log Minkowski inequality from concavity, and in Theorem 1.2 to connect maximizers of the variational functional to solutions of the measure equation.
  • standard math Regularity theorem for the dual Minkowski problem (Böröczky-Fodor [2, Theorem 1.5])
    Used in Remark 4.2 to upgrade the maximizer to a smooth solution when the measure has a smooth positive density.

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Pith. "Pith review of The uniqueness and non-uniqueness of solutions to the even dual Minkowski problem." pith.science (2026). https://pith.science/paper/3P54GXY5

@misc{pith2026260801912,
  author       = {Pith},
  title        = {Pith review of: The uniqueness and non-uniqueness of solutions to the even dual Minkowski problem},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/3P54GXY5}},
  note         = {Machine review of arXiv:2608.01912}
}
abstract

For solutions to the Minkowski problem of the even dual curvature measures $\widetilde{C}_q$ in $\mathbb{R}^n$, $n\ge 2$, we prove the non-uniqueness for $q>n$ and $n\ge 2$, and the uniqueness for $0<q<n$ and $n=2$. These results are governed by our established logarithmic Brunn-Minkowski inequality for dual quermassintegrals $\widetilde V_q$.

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Cited by 1 Pith paper

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

  1. On the Brunn-Minkowski inequality for $q$-th dual quermassintegrals with $q>n$

    math.DG 2026-08 conditional novelty 7.0 of 10

    For dual quermassintegrals, the Brunn-Minkowski inequality fails for q>n (and q>n+2 for origin-symmetric bodies), holds at q=n+2 for origin-symmetric bodies, and holds for 0<q≤n+1 for unconditional bodies.

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Works this paper leans on

38 extracted references · 35 canonical work pages · cited by 1 Pith paper

  1. [5]

    B¨ or¨ oczky, E

    K. B¨ or¨ oczky, E. Lutwak, D. Yang, G. Zhang, The log-Brunn–Minkowski inequality, Adv. Math. 231 (2012) 1974–1997. 2, 3, 9

    work page 2012

  2. [2]

    B¨ or¨ oczky, F

    K. B¨ or¨ oczky, F. Fodor, TheLp dual Minkowski problem forp >1 andq >0, J. Differential Equations 266 (2019) 7980–8033. 6

    work page 2019

  3. [1]

    Aleksandrov, Existence and uniqueness of a convex surface with a given integral curvature, C

    A. Aleksandrov, Existence and uniqueness of a convex surface with a given integral curvature, C. R. (Doklady) Acad. Sci. URSS (N.S.) 35 (1942) 131–134. 2

    work page 1942

  4. [3]

    B¨ or¨ oczky, M

    K. B¨ or¨ oczky, M. Henk, H. Pollehn, Subspace concentration of dual curvature measures of symmetric convex bodies, J. Differential Geom. 109 (2018) 411–429. 2

    work page 2018

  5. [4]

    B¨ or¨ oczky, P

    K. B¨ or¨ oczky, P. Kalantzopoulos, Log-Brunn–Minkowski inequality under symmetry, Trans. Amer. Math. Soc. 375 (2022) 5987–6013. 2, 4, 5

    work page 2022

  6. [6]

    B¨ or¨ oczky, E

    K. B¨ or¨ oczky, E. Lutwak, D. Yang, G. Zhang, The logarithmic Minkowski problem, J. Amer. Math. Soc. 26 (2013) 831–852. 2

    work page 2013

  7. [7]

    B¨ or¨ oczky, E

    K. B¨ or¨ oczky, E. Lutwak, D. Yang, G. Zhang, Y. Zhao, The dual Minkowski problem for symmetric convex bodies, Adv. Math. 356 (2019) Paper No. 106805. 2

    work page 2019

  8. [8]

    B¨ or¨ oczky, S

    K. B¨ or¨ oczky, S. Chen, W. Liu, C. Saroglou, Uniqueness in the near isotropicLp dual Minkowski problem, arXiv:2505.01066, 2025. 2

    arXiv 2025

Show all 38 references
  1. [9]

    H. Chen, S. Chen, Q. Li, Variations of a class of Monge–Amp` ere-type functionals and their applications, Anal. PDE 14 (2021) 689–716. 3

    2021

  2. [10]

    S. Chen, Y. Huang, Q.-R. Li, J. Liu, TheL p-Brunn–Minkowski inequality forp <1, Adv. Math. 368 (2020) Paper No. 107166. 2, 4

    2020

  3. [11]

    C. Chen, Y. Huang, Y. Zhao, Smooth solutions to theL p dual Minkowski problem, Math. Ann. 373 (2019) 953–976. 2

    2019

  4. [12]

    S. Chen, Q. Li, G. Zhu, The logarithmic Minkowski problem for non-symmetric measures, Trans. Amer. Math. Soc. 371 (2019) 2623–2641. 3

    2019

  5. [13]

    Colesanti, G

    A. Colesanti, G. Livshyts, A. Marsiglietti, On the stability of Brunn–Minkowski type inequalities, J. Funct. Anal. 273 (2017) 1120–1139. 4 15

    2017

  6. [14]

    Gardner, Geometric tomography, 2nd ed., Cambridge University Press, Cambridge, 2006

    R. Gardner, Geometric tomography, 2nd ed., Cambridge University Press, Cambridge, 2006. 4

    2006

  7. [15]

    Gardner, The Brunn–Minkowski inequality, Bull

    R. Gardner, The Brunn–Minkowski inequality, Bull. Amer. Math. Soc. 39 (2002) 355–405. 5

    2002

  8. [16]

    Gilbarg, N

    D. Gilbarg, N. Trudinger, Elliptic partial differential equations of second order, 2nd ed., Springer-Verlag, Berlin, 2001. 6

    2001

  9. [17]

    Y. He, Q. Li, X. Wang, Multiple solutions of theL p-Minkowski problem, Calc. Var. Partial Differential Equations 55 (2016) Paper No. 117. 3

    2016

  10. [18]

    Hu, The dual Minkowski problem for positive indices, Int

    J. Hu, The dual Minkowski problem for positive indices, Int. Math. Res. Not. IMRN 2025 (2025) Paper No. rnaf192. 2

    2025

  11. [19]

    Y. Hu, M. Ivaki, Weighted centro-affine Poincar´ e inequalities, arXiv:2606.04774, 2026. 2

    2026 arXiv

  12. [20]

    Huang, Y

    Y. Huang, Y. Jiang, Variational characterization for the planar dual Minkowski problem, J. Funct. Anal. 277 (2019) 2209–2236. 3

    2019

  13. [21]

    Huang, E

    Y. Huang, E. Lutwak, D. Yang, G. Zhang, Geometric measures in the dual Brunn–Minkowski theory and their associated Minkowski problems, Acta Math. 216 (2016) 325–388. 1, 2, 5, 6

    2016

  14. [22]

    Ivaki, E

    M. Ivaki, E. Milman, TheL p-Minkowski problem under curvature pinching, Int. Math. Res. Not. IMRN 2024 (2024) 8638–8652. 2, 4

    2024

  15. [23]

    H. Jian, J. Lu, X. Wang, Nonuniqueness of solutions to theL p-Minkowski problem, Adv. Math. 281 (2015) 845–856. 3

    2015

  16. [24]

    Kolesnikov, E

    A. Kolesnikov, E. Milman, LocalL p-Brunn–Minkowski inequalities forp <1, Mem. Amer. Math. Soc. 277 (2022), no. 1360, v+78 pp. 2, 4

    2022

  17. [25]

    Q. Li, J. Liu, J. Lu, Nonuniqueness of solutions to theL p dual Minkowski problem, Int. Math. Res. Not. IMRN 2022 (2022) 9114–9150. 3

    2022

  18. [26]

    Li, Nonuniqueness of solutions to theL p chord Minkowski problem, Calc

    Y. Li, Nonuniqueness of solutions to theL p chord Minkowski problem, Calc. Var. Partial Differential Equa- tions 63 (2024) Paper No. 83. 3

    2024

  19. [27]

    Lutwak, Dual mixed volumes, Pacific J

    E. Lutwak, Dual mixed volumes, Pacific J. Math. 58 (1975) 531–538. 1

    1975

  20. [28]

    Milman, Centro-affine differential geometry and the log-Minkowski problem, J

    E. Milman, Centro-affine differential geometry and the log-Minkowski problem, J. Eur. Math. Soc. 27 (2025) 709–772. 2, 4

    2025

  21. [29]

    Rotem, A letter: the log-Brunn–Minkowski inequality for complex bodies, arXiv:1412.5321, 2014

    L. Rotem, A letter: the log-Brunn–Minkowski inequality for complex bodies, arXiv:1412.5321, 2014. 3

    2014 arXiv

  22. [30]

    Saroglou, Remarks on the conjectured log-Brunn–Minkowski inequality, Geom

    C. Saroglou, Remarks on the conjectured log-Brunn–Minkowski inequality, Geom. Dedicata 177 (2015) 353–

    2015

  23. [31]

    Schneider, Convex bodies: the Brunn–Minkowski theory, Cambridge University Press, 2014

    R. Schneider, Convex bodies: the Brunn–Minkowski theory, Cambridge University Press, 2014. 4

    2014

  24. [32]

    Stancu, The discrete planarL 0-Minkowski problem, Adv

    A. Stancu, The discrete planarL 0-Minkowski problem, Adv. Math. 167 (2002) 160–174. 2

    2002

  25. [33]

    Stancu, On the number of solutions to the discrete two-dimensionalL 0-Minkowski problem, Adv

    A. Stancu, On the number of solutions to the discrete two-dimensionalL 0-Minkowski problem, Adv. Math. 180 (2003) 290–323. 2

    2003

  26. [34]

    van Handel, The local logarithmic Brunn–Minkowski inequality for zonoids, Geometric aspects of func- tional analysis, Lecture Notes in Mathematics, 2327, Springer, Cham, 2023, pp

    R. van Handel, The local logarithmic Brunn–Minkowski inequality for zonoids, Geometric aspects of func- tional analysis, Lecture Notes in Mathematics, 2327, Springer, Cham, 2023, pp. 355–379. 4

    2023

  27. [35]

    Wang, Monotonicity of the deficit in the local log-Brunn–Minkowski inequality, arXiv:2510.19061, 2025

    S. Wang, Monotonicity of the deficit in the local log-Brunn–Minkowski inequality, arXiv:2510.19061, 2025. 4

    2025

  28. [36]

    Xi, The reverse-log-Brunn–Minkowski inequality, arXiv:2307.04266, 2023

    D. Xi, The reverse-log-Brunn–Minkowski inequality, arXiv:2307.04266, 2023. 4

    2023 arXiv

  29. [37]

    Zhao, Existence of solutions to the even dual Minkowski problem, J

    Y. Zhao, Existence of solutions to the even dual Minkowski problem, J. Differential Geom. 110 (2018) 543–

    2018

  30. [38]

    Zhao, The dual Minkowski problem for negative indices, Calc

    Y. Zhao, The dual Minkowski problem for negative indices, Calc. Var. Partial Differential Equations 56 (2017) Paper No. 18. 2 16

    2017

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