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arxiv: 2604.10924 · v1 · submitted 2026-04-13 · 🧮 math.AP

The L_p dual Christoffel-Minkowski type problem for a class of Hessian quotient equations

Shasha Luo , Jiabao Gong , Qiang Tu This is my paper

Pith reviewed 2026-05-10 16:27 UTC · model grok-4.3

classification 🧮 math.AP
keywords Hessian quotientdual Christoffel-Minkowski probleminverse convexityfull rank theoremspherically convexexistence uniquenessa priori estimatesfully nonlinear PDE
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The pith

Existence and uniqueness of strictly spherically convex solutions are proven for the Lp dual Christoffel-Minkowski type problem using Hessian quotient operators.

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

This paper studies an Lp dual version of the Christoffel-Minkowski problem for the Hessian quotient operator, which is the ratio of two elementary symmetric polynomials applied to a matrix Lambda. The authors use a recently found inverse convexity property to prove a full rank theorem when certain structural assumptions are met. With the help of a priori estimates, they then show that strictly spherically convex solutions to the problem both exist and are unique. This matters because it provides a general existence result for a family of fully nonlinear equations that arise when prescribing dual curvature measures in convex geometry.

Core claim

Exploiting the inverse convexity property of the Hessian quotient operators, the paper establishes a full rank theorem under suitable structural assumptions. Together with a priori estimates, this result proves the existence and uniqueness of strictly spherically convex solutions to the Lp dual Christoffel-Minkowski type problem.

What carries the argument

Inverse convexity property of the Hessian quotient operator sigma_k(Lambda)/sigma_l(Lambda), used to obtain the full rank theorem.

Load-bearing premise

The Hessian quotient operator satisfies the inverse convexity property under the structural assumptions, allowing the full rank theorem and the application of a priori estimates to the dual problem.

What would settle it

A counterexample consisting of a Hessian quotient operator satisfying the structural assumptions for which the associated PDE has either no solution or multiple strictly spherically convex solutions would falsify the result.

read the original abstract

In this paper, we investigate an $L_p$ dual Christoffel-Minkowski type problem for the Hessian quotient operator $\frac{\sigma_{k}(\Lambda)}{\sigma_{l}(\Lambda)}$, where the operator $\Lambda$ has been widely studied in the literature. Exploiting the recently discovered ``inverse convexity'' property of this class of operators, we establish a full rank theorem under suitable structural assumptions. Together with a priori estimates, this result enables us to prove the existence and uniqueness of strictly spherically convex solutions to the above $L_p$ dual Christoffel-Minkowski type problem.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 1 minor

Summary. The manuscript investigates the L_p dual Christoffel-Minkowski type problem for the Hessian quotient operator σ_k(Λ)/σ_l(Λ). Exploiting the recently discovered inverse convexity property of this class of operators, the authors establish a full rank theorem under suitable structural assumptions. Combined with a priori estimates, this is used to prove existence and uniqueness of strictly spherically convex solutions.

Significance. If the structural assumptions are verified to hold in the L_p dual setting, the work would extend the existence theory for convex solutions of Hessian quotient equations to their dual counterparts, building directly on the inverse convexity property. This methodological link is a strength, as it avoids parameter-fitting and relies on the cited external property to obtain the full-rank conclusion needed for the existence result.

major comments (1)
  1. [statement of the full rank theorem and the main existence theorem] The full rank theorem (invoked to reach existence/uniqueness) rests on the inverse convexity property holding under 'suitable structural assumptions' for the specific L_p dual equation. The dual formulation replaces the standard measure with an L_p-weighted one, which can alter the monotonicity or sign conditions required for inverse convexity. The manuscript must explicitly verify that the assumptions (e.g., admissible range of k,l and derivative signs) remain satisfied; without this check the transfer of the full-rank result is not justified.
minor comments (1)
  1. [Introduction and abstract] Clarify the precise range of k and l for which the structural assumptions are claimed to hold, and ensure all citations to the inverse-convexity reference are complete.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the positive summary and for identifying a point that requires clarification. We address the major comment below and will revise the manuscript accordingly.

read point-by-point responses

  1. Referee: The full rank theorem (invoked to reach existence/uniqueness) rests on the inverse convexity property holding under 'suitable structural assumptions' for the specific L_p dual equation. The dual formulation replaces the standard measure with an L_p-weighted one, which can alter the monotonicity or sign conditions required for inverse convexity. The manuscript must explicitly verify that the assumptions (e.g., admissible range of k,l and derivative signs) remain satisfied; without this check the transfer of the full-rank result is not justified.

    Authors: We agree that an explicit verification strengthens the argument. The inverse convexity property is a structural feature of the Hessian quotient operator itself and does not depend on the choice of measure (standard or L_p-weighted). Consequently the admissible ranges for k and l (1 ≤ l < k ≤ n) and the required sign conditions on the first and second derivatives of the operator remain identical to those in the non-dual setting. In the revised manuscript we will add a short paragraph immediately after the statement of the full-rank theorem that records this verification, confirming that all structural hypotheses continue to hold for the L_p dual Christoffel-Minkowski problem. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation relies on external property and standard estimates

full rationale

The paper's central chain invokes the inverse convexity property of the Hessian quotient operator as a recently discovered external fact, then combines it with a priori estimates to obtain a full-rank theorem and existence/uniqueness for the Lp dual problem. No self-definitional reduction, fitted-input prediction, or load-bearing self-citation chain is present; the structural assumptions are stated as external inputs rather than derived from the target result itself. The argument remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 3 axioms · 0 invented entities

The central claim rests on standard background results in convex geometry and fully nonlinear PDE theory plus the recently discovered inverse convexity property; no free parameters are fitted to data, and no new entities are postulated.

axioms (3)
  • domain assumption The Hessian quotient operator satisfies the inverse convexity property under suitable structural assumptions on k, l, and the data.
    Invoked explicitly to establish the full rank theorem; this is presented as recently discovered rather than proved in the paper.
  • domain assumption A priori estimates hold for the solutions to the Lp dual problem.
    Used together with the full rank theorem to conclude existence and uniqueness; details not provided in abstract.
  • standard math Standard ellipticity and convexity conditions for the fully nonlinear operator in the geometric setting.
    Background assumptions typical for such existence results in Hessian equations.

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Works this paper leans on

64 extracted references · 64 canonical work pages

  1. [1]

    Aleksandrov, On the theory of mixed volumes

    A. Aleksandrov, On the theory of mixed volumes. III. Extensions of two theorems of Minkowski on convex polyhedra to arbitrary convex bodies, Mat. Sb. (N.S.) 3, 27-46, (1938)

    work page 1938

  2. [2]

    Aleksandrov, On the surface area measure of convex bodies, Mat

    A. Aleksandrov, On the surface area measure of convex bodies, Mat. Sb. (N.S.) 6, 167-174, (1939)

    work page 1939

  3. [3]

    Math., 177, 203-335, (2009)

    B.Bian, P.Guan, A microscopic convexity principle for nonlinear partial differential equations, Invent. Math., 177, 203-335, (2009)

    work page 2009

  4. [4]

    Berg, Corps convexes et potentiels sph´ eriques, Mat.-Fys

    C. Berg, Corps convexes et potentiels sph´ eriques, Mat.-Fys. Medd. Danske Vid. Selsk., 37(6), 64, (1969)

    work page 1969

  5. [5]

    B¨ or¨ oczky, F

    K. B¨ or¨ oczky, F. Fodor, TheLp dual Minkowski problem forp >1 andq >0, J. Differential Equations, 266(12), 7980-8033, (2019)

    work page 2019

  6. [6]

    Bianchi, K

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

    work page 2019

  7. [7]

    Bryan, M

    P. Bryan, M. Ivaki, J. Scheuer, Christoffel-Minkowski flows, Trans. Amer. Math. Soc., 376(4), 2373-2393, (2023)

    work page 2023

  8. [8]

    B¨ or¨ oczky, E

    K. B¨ or¨ oczky, E. Lutwak, D. Yang, G. Zhang, The logarithmic Minkowski problem, J. Amer. Math. Soc., 26(3), 831-852, (2013)

    work page 2013

  9. [9]

    B¨ or¨ oczky, H

    K. B¨ or¨ oczky, H. T. Trinh, The planarLp-Minkowski problem for 0< p <1, Adv. in Appl. Math., 87, 58-81, (2017)

    work page 2017

  10. [10]

    Caffarelli, L

    L. Caffarelli, L. Nirenberg, J. Spruck, The Dirichlet problem for nonlinear second order elliptic equations, III: Functions of the eigenvalues of the Hessian, Acta Math., 155, 261-301, (1985)

    work page 1985

  11. [11]

    Cabezas-Moreno, J

    C. Cabezas-Moreno, J. Hu, TheLpdual Christoffel-Minkowski problem for 1< p < q≤k+ 1 with 1≤k≤n, Calc. Var. Partial Differential Equations, 64(7), 229, (2025)

    work page 2025

  12. [12]

    Chen, On the elementary symmetric functions, preprint

    C. Chen, On the elementary symmetric functions, preprint

    work page

  13. [13]

    C. Chen, W. Dong, F. Han, Interior Hessian estimates for a class of Hessian type equations, Calc. Var. Partial Differential Equations, 62(52), 1-15, (2023)

    work page 2023

  14. [14]

    C. Chen, L. Xu, TheL p Minkowski type problem for a class of mixed Hessian quotient equations, Adv. Math., 411, 108794, (2022)

    work page 2022

  15. [15]

    C. Chen, L. Xu, Uniqueness of solutions to a class of mixed Hessian quotient type equations, J. Geom. Anal., 33(210), 18, (2023) Lp DUAL CHRISTOFFEL-MINKOWSKI TYPE PROBLEM 29

    work page 2023

  16. [16]

    C. Chen, Y. Huang, Y. Zhao, Smooth solutions to theL p dual Minkowski problem, Math. Ann., 373(3-4), 953-976, (2019)

    work page 2019

  17. [17]

    X. Chen, Q. Tu, N. Xiang, A class of Hessian quotient equations in Euclidean space, J. Differential Equations, 269(2020), 11172-11194, (2020)

    work page 2020

  18. [18]

    X. Chen, Q. Tu, N. Xiang, The Dirichlet problem for a class of Hessian quotient equa- tions on Riemannian manifolds, Int. Math. Res. Not., 2023(12), 10013-10036, (2023)

    work page 2023

  19. [19]

    X. Chen, Q. Tu, N. Xiang, TheL p dual Christoffel-Minkowski problem for the case p≥q, arXiv: 2503.01454 (2025)

    work page arXiv 2025

  20. [20]

    L. Chen, Q. Tu, N. Xiang, Pogorelov type estimates for a class of Hessian quotient equations, J. Differential Equations, 282, 272-284, (2021)

    work page 2021

  21. [21]

    S. Chen, Q. Li, G. Zhu, On theL p Monge-Amp´ ere equation, J. Differential Equations, 263(8), 4997-5011, (2017)

    work page 2017

  22. [22]

    H. Chen, Q. Li, TheL p dual Minkowski problem and related parabolic flows, J. Funct. Anal., 281(8), 109139, (2021)

    work page 2021

  23. [23]

    Cheng, S

    S. Cheng, S. Yau, On the regularity of the solution of then-dimensionalMinkowski problem, Comm. Pure Appl. Math., 29(5), 495-516, (1976)

    work page 1976

  24. [24]

    Christoffel, Ueber die Bestimmung der Gestalt einer krummen Oberflche durch lokale Messungenauf derselben, J

    E. Christoffel, Ueber die Bestimmung der Gestalt einer krummen Oberflche durch lokale Messungenauf derselben, J. Reine Angew. Math., 64, 193-209, (1865)

    work page

  25. [25]

    K. Chou, X. Wang, TheL p-Minkowski problem and the Minkowski problem in cen- troaffine geometry, Adv. Math., 205, 33-83, (2006)

    work page 2006

  26. [26]

    J. Chu, H. Jiao, Curvature estimates for a class of Hessian type equations, Calc. Var. Partial Differential Equations, 60(90), 1-18, (2021)

    work page 2021

  27. [27]

    Dong, Curvature estimates for p-convex hypersurfaces of prescribed curvature, Rev

    W. Dong, Curvature estimates for p-convex hypersurfaces of prescribed curvature, Rev. Mat. Iberoam., 39, 1039-1058, (2023)

    work page 2023

  28. [28]

    Dong, The Dirichlet problem for prescribed curvature equations of p-convex hyper- surfaces, Manuscripta Math., 174, 785-806, (2024)

    W. Dong, The Dirichlet problem for prescribed curvature equations of p-convex hyper- surfaces, Manuscripta Math., 174, 785-806, (2024)

    work page 2024

  29. [29]

    Dinew, Interior estimates for p-plurisubharmonic functions, Indiana Univ

    S. Dinew, Interior estimates for p-plurisubharmonic functions, Indiana Univ. Math. J., 72(5), 2025-2057, (2023)

    work page 2025

  30. [30]

    S. Ding, G. Li, A class of inverse curvature flows andL p dual Christoffel-Minkowski problem, Trans. Amer. Math. Soc., 376(1), 697-752, (2023)

    work page 2023

  31. [31]

    Firey, The determination of convex bodies from their mean radius of curvature functions, Mathematika, 14, 1-13, (1967)

    W. Firey, The determination of convex bodies from their mean radius of curvature functions, Mathematika, 14, 1-13, (1967)

    work page 1967

  32. [32]

    Firey, Christoffel’s problem for general convex bodies, Mathematika, 15, 7-21, (1968)

    W. Firey, Christoffel’s problem for general convex bodies, Mathematika, 15, 7-21, (1968)

    work page 1968

  33. [33]

    Firey, Intermediate Christoffel-Minkowski problems for figures of revolution, Israel J

    W. Firey, Intermediate Christoffel-Minkowski problems for figures of revolution, Israel J. Math., 8, 384-390, (1970)

    work page 1970

  34. [34]

    Gauduchon, La 1-forme de torsion d’une variete hermitienne compacte, Math

    P. Gauduchon, La 1-forme de torsion d’une variete hermitienne compacte, Math. Ann., 267, 495-518, (1984)

    work page 1984

  35. [35]

    Gerhardt, Curvature problems, Series in Geometry and Topology, International Press of Boston Inc

    C. Gerhardt, Curvature problems, Series in Geometry and Topology, International Press of Boston Inc. Sommerville, 39, (2006)

    work page 2006

  36. [36]

    B. Guan, X. Nie, Second order estimates for fully nonlinear elliptic equations with gradient terms on Hermitian manifolds, arXiv :2108 .03308

    work page

  37. [37]

    J. Gong, Z. Liu, Q. Tu, The Neumann problem for a class of Hessian quotient type equations, J. Differential Equations, 431, 113251, (2025) 30 SHASHA LUO, JIABAO GONG, AND QIANG TU ∗

    work page 2025

  38. [38]

    Guan, Topics in geometric fully nonlinear equations, Lecture Notes, (2002)

    P. Guan, Topics in geometric fully nonlinear equations, Lecture Notes, (2002)

    work page 2002

  39. [39]

    P. Guan, X. Ma, The Christoffel-Minkowski problem. I. Convexity of solutions of aHes- sian equation, Invent. Math, 151(3), 553-577, (2003)

    work page 2003

  40. [40]

    P. Guan, C. Lin, On equation det(u ij +δ iju) =u pfonS n, manuscript, (1999)

    work page 1999

  41. [41]

    P. Guan, C. Lin, X. Ma, The Christoffel-Minkowski problem. II.Weingarten curvature equations, Chinese Ann. Math. Ser. B, 27(6), 595-614, (2006)

    work page 2006

  42. [42]

    P. Guan, X. Ma, F. Zhou, The Christofel-Minkowski problem. III. Existence and con- vexity of admissible solutions, Comm. Pure Appl. Math., 59(9), 1352-1376, (2006)

    work page 2006

  43. [43]

    P. Guan, C. Xia,L p Christoffel-Minkowski problem: the case 1< p < k+ 1, Calc. Var. Partial Differential Equations, 57 (2018): 69

    work page 2018

  44. [44]

    Math., 138, 151-161, (1999)

    Gauss curvature flow, the fate of the rolling stones, Invent. Math., 138, 151-161, (1999)

    work page 1999

  45. [45]

    Harvey, H

    F. Harvey, H. Lawson, p-convexity, p-plurisubharmonicity and the Levi problem, Indi- ana Univ. Math. J., 62, 149-169, (2013)

    work page 2013

  46. [46]

    C. Hu, X. Ma, C. Shen, On the Christoffel-Minkowski problem of Firey’sp-sum, Calc. Var. Partial Differential Equations, 21(2), 137-155, (2004)

    work page 2004

  47. [47]

    Huang, Y

    Y. Huang, Y. Zhao, On theL p dual Minkowski problem, Adv. Math., 332, 57-84, (2018)

    work page 2018

  48. [48]

    Ivaki, Deforming a hypersurface by principal radii of curvature and support function, Calc

    M. Ivaki, Deforming a hypersurface by principal radii of curvature and support function, Calc. Var. Partial Differential Equations, 58(1), 1, (2019)

    work page 2019

  49. [49]

    Lieberman, Second order parabolic differential equations, World Scientific, (1996)

    G. Lieberman, Second order parabolic differential equations, World Scientific, (1996)

    work page 1996

  50. [50]

    J. Lu, X. Wang, Rotationally symmetric solutions to theL p-Minkowski problem, J. Differential Equations, 254(3), 983-1005, (2013)

    work page 2013

  51. [51]

    Lutwak, The Brunn-Minkowski-Firey theory

    E. Lutwak, The Brunn-Minkowski-Firey theory. I. Mixed volumes and the Minkowski problem, J. Differential Geom., 38(1), 131-150, (1993)

    work page 1993

  52. [52]

    Lutwak, D

    E. Lutwak, D. Yang, G. Zhang, On theL p-Minkowski problem, Trans. Amer. Math. Soc., 356(11), 4359-4370, (2004)

    work page 2004

  53. [53]

    Lutwak, D

    E. Lutwak, D. Yang, G. Zhang,L p dual curvature measures, Adv. Math., 329, 85-132, (2018)

    work page 2018

  54. [54]

    Minkowski, Allgemeine Lehrs¨ atze ¨ uber die convexen Polyeder, Nachr

    H. Minkowski, Allgemeine Lehrs¨ atze ¨ uber die convexen Polyeder, Nachr. Ges. Wiss. Gttingen, 198-219, (1897)

    work page

  55. [55]

    Minkowski, Volumen und Oberfl¨ ache, Math

    H. Minkowski, Volumen und Oberfl¨ ache, Math. Ann., 57(4), 447-495, (1903)

    work page 1903

  56. [56]

    Nirenberg, The Weyl and Minkowski problems in differential geometry in the large, Comm

    L. Nirenberg, The Weyl and Minkowski problems in differential geometry in the large, Comm. Pure Appl. Math., 6(3), 337-394, (1953)

    work page 1953

  57. [57]

    Pogorelov, The Minkowski multidimensional problem, translated from the Russian by Vladimir Oliker,Scripta Series in Mathematics, Winston, Washington, DC (1978)

    A. Pogorelov, The Minkowski multidimensional problem, translated from the Russian by Vladimir Oliker,Scripta Series in Mathematics, Winston, Washington, DC (1978)

    work page 1978

  58. [58]

    Sheng, C

    W. Sheng, C. Yi, A class of anisotropic expanding curvature flows, Discrete Contin. Dyn. Syst., 40(4), 2017-2035, (2020)

    work page 2017

  59. [59]

    Sz´ ekelyhidi, V

    G. Sz´ ekelyhidi, V. Tosatti, B. Weinkove, Gauduchon metrics with prescribed volume form, Acta Math., 219, 181-211, (2017)

    work page 2017

  60. [60]

    Tosatti, B

    V. Tosatti, B. Weinkove, The Monge-Amp´ ere equation for (n−1)-plurisubharmonic functions on a compact K¨ ahler manifold, J. Amer. Math. Soc., 30(2), 311-346, (2017)

    work page 2017

  61. [61]

    Zhu, The logarithmic Minkowski problem for polytopes, Adv

    G. Zhu, The logarithmic Minkowski problem for polytopes, Adv. Math., 262, 909-931, (2014)

    work page 2014

  62. [62]

    Zhu, TheL p-Minkowski problem for polytopes for 0< p <1, J

    G. Zhu, TheL p-Minkowski problem for polytopes for 0< p <1, J. Funct. Anal., 269(4), 1070-1094, (2015) Lp DUAL CHRISTOFFEL-MINKOWSKI TYPE PROBLEM 31

    work page 2015

  63. [63]

    Zhu, The centro-affine Minkowski problem for polytopes, J

    G. Zhu, The centro-affine Minkowski problem for polytopes, J. Differential Geom., 101(1), 159-174, (2015)

    work page 2015

  64. [64]

    Zhou, Curvature estimates for a class of Hessian quotient type curvature equations, Calc

    J. Zhou, Curvature estimates for a class of Hessian quotient type curvature equations, Calc. Var. Partial Differential Equations, 63(4), 88, (2024) F aculty of Mathematics and Statistics, Hubei Key Laboratory of Applied Mathe- matics, Hubei University, Wuhan 430062, P.R. China Email address:202421104011295@stu.hubu.edu.cn F aculty of Mathematics and Stati...

    work page 2024