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REVIEW 3 major objections 3 minor 64 references

The capillary Orlicz-Minkowski problem

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Pith's one-line read This paper proves existence of smooth symmetric volume-one solutions to the capillary Orlicz-Minkowski problem, with equality in the lower bound forcing a spherical cap.

desk verdict Natural capillary Orlicz theory with solid inequalities, but the main existence theorem has a missing starting point: the t=0 solution is not in the solution space H. read the letter →

arxiv 2509.10859 v1 pith:BQ2OJ66Y submitted 2025-09-13 math.DG

Xudong Wang , Baocheng Zhu This is my paper · ORCID
classification math.DG MSC 53C4253C4535J66
keywords capillaryhypersurfaceOrlicz-MinkowskiproblemRobinboundaryvalueconditionOrlicz-Brunn-MinkowskiinequalityMonge-Ampereequationcontinuitymethodsphericalcapconvexgeometry
topics The Measurement Problem
open problems The Measurement Problem
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 introduces a Robin-boundary analogue of the Orlicz-Minkowski problem: it asks for a convex body meeting the upper half-space at a fixed contact angle whose capillary Orlicz surface area measure is a prescribed positive even function on the spherical cap. The main theorem asserts that for every admissible weight function, satisfying mild growth and convexity conditions, and every even positive datum satisfying an integral lower bound, there is a smooth symmetric capillary convex body of volume one whose support function solves the associated Monge-Ampere equation with Robin boundary condition. If the lower bound is an equality and the weight is strictly convex, the only solution is the spherical cap. This matters because it extends the classical Minkowski and Orlicz-Minkowski problems to the capillary setting and, as a corollary, yields volume-normalized supercritical capillary Lp solutions with the normalization constant forced to 1.

What carries the argument

The load-bearing object is the capillary Orlicz surface area measure $dS^c_\phi(b\Sigma,\xi)=\phi(\ell/h)\,h\,\det(h_{ij}+h\delta_{ij})\,d\xi$ on the spherical cap $C_\theta$, which converts the geometric prescription into a Robin-boundary Monge-Ampere equation. The proof mechanism is the one-parameter continuity family with $f_t=(1-t)\phi(1)\ell+tf$; the solution space $H$ is defined by the orthogonality condition (1.3), which makes the linearized operator $L_h$ self-adjoint and surjective on $H$ through Lemmas 5.1 and 5.2. The a priori estimates of Section 4 close the method: the $C^0$ estimate uses the growth of $\phi$ and the Steiner point argument for even functions, the $C^1$ estimate uses an auxiliary function with the distance-to-boundary term, the $C^2$ estimate uses maximum principles on $P=\nabla^2_{\Xi,\Xi}h+h$ and a boundary auxiliary function $Q$, and bootstrap regularity then gives all higher norms. On the geometric side, the Orlicz-Minkowski inequality follows from Jensen's inequality together with the Aleksandrov-Fenchel inequality, and it supplies the equality case identifying the spherical cap.

What would settle it

Compute the orthogonality integral for $h=\ell$ with $v=\ell-\bar{\ell}$, where $\bar{\ell}$ is the average of $\ell$ over $C_\theta$. When $\phi'(1)\neq(n+1)\phi(1)$, the premise in (1.3) holds because $\int v=0$, yet $\int \ell v=\int(\ell-\bar{\ell})^2>0$, so the orthogonality condition fails; when $\phi'(1)=(n+1)\phi(1)$, the premise holds for every $v$, and taking $v=\ell$ gives $\int \ell^2>0$. Thus $h=\ell$ is not in $H$, and this single calculation decides whether the continuity method has a starting solution.

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Extended reading notes

Core claim

On the paper's own terms, the central discovery is that the prescribed-capillary-Orlicz-measure problem is solvable in the smooth category. Writing $h$ for the capillary support function on $C_\theta=\{\xi\in\mathbb{R}^{n+1}_+: |\xi-\cos\theta\,e|=1\}$, with $\ell(\xi)=\sin^2\theta+\cos\theta\langle\xi,e\rangle$, the problem is the Robin-boundary Monge-Ampere equation $$\$\varphi$(\ell/h)\,h\,\det(h_{ij}+h\delta_{ij})=f\ \text{in }C_\$\theta$,\qquad \nabla_\mu h=\cot\$\theta$\,h\ \text{on }\partial C_\$\theta$.$$ Theorem 1.1 asserts that for every $\phi\in\mathcal{O}$ and every even positive $f\in C^2(C_\theta)$ satisfying the lower bound (1.4), this equation has a smooth, symmetric solution of volume one which also satisfies the orthogonality condition with respect to $f$; if equality holds in (1.4) and $\phi$ is strictly convex, the solution is the spherical cap $|bC_\theta|^{-1/(n+1)}C_\theta$. The proof is a continuity method in the interpolating family $f_t=(1-t)\phi(1)\ell+tf$: a priori $C^0,C^1,C^2$ and higher estimates control solutions, the linearized operator is symmetric on the constrained space $H$, its kernel is characterized by Lemma 5.2, and the implicit function theorem gives openness, making the solvable set both open and closed. The companion capillary Orlicz-Minkowski and Orlicz-Brunn-Minkowski inequalities carry the equality and rigidity part.

Load-bearing premise

The argument's load-bearing premise is that the spherical cap solution $h=\ell$ belongs to the constrained solution space $H$ used for the continuity method; that condition rules $h=\ell$ out, so the base case at $t=0$ is not actually established.

Editorial extensions

If this is right

  • For $\phi(x)=x^p$ with $p\ge n+1$, Theorem 1.1 gives smooth symmetric volume-one solutions to the capillary even $L^p$-Minkowski problem; in the critical case $p=n+1$ the normalization constant $\gamma$ from [46] is forced to be 1.
  • Equality in the lower bound (1.4) with strictly convex $\phi$ pins the solution uniquely: the spherical cap $|bC_\theta|^{-1/(n+1)}C_\theta$ is the only volume-one capillary Orlicz solution.
  • The capillary Orlicz-Brunn-Minkowski inequality (3.4) makes capillary Orlicz addition a subadditive operation on volumes, with equality precisely for dilates when $\phi$ is strictly convex.
  • The a priori estimates in Lemmas 4.1-4.5 control solutions in every $C^{m+1,\alpha}$ norm by the data, so higher-regularity and compactness statements follow for the same class of Robin Monge-Ampere data.

Reading between the lines

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

  • The base step of the continuity method is not verified as written: at $h=\ell$ and $t=0$, the premise in (1.3) holds for every $v$ with $\int_{C_\theta} v=0$ (or for every $v$ when $\phi'(1)=(n+1)\phi(1)$), while $\int_{C_\theta}\ell v$ need not vanish, so $\ell\notin H$ and the parameter set $I$ may be empty at $t=0$.
  • Because the only role of $H$ is to make the linearized operator $L_h$ surjective, a transversality condition that $\ell$ does satisfy could replace the orthogonality condition and likely restore the base step while preserving the openness argument.
  • The evenness assumption drives the $C^0$ estimate by forcing the Steiner point to the origin; non-even data would require a different normalization or a translation argument, so the theorem as stated does not cover them.
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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 / 3 minor

Summary. The paper introduces a Robin-boundary analogue of the Orlicz-Minkowski problem, called the capillary Orlicz-Minkowski problem. For a convex function φ in a class O and a positive even function f on a spherical cap Cθ, the problem asks for a symmetric capillary convex body of volume one whose support function h solves φ(ℓ/h)h det(hij+hδij)=f in Cθ with Robin condition ∇μh=cotθ h on ∂Cθ. The authors define a capillary Orlicz surface area measure, establish capillary Orlicz-Minkowski and Orlicz-Brunn-Minkowski inequalities using the Alexandrov-Fenchel inequality from [48], derive a priori estimates for a normalized version of the equation, and then prove the main existence theorem by a continuity method, with a uniqueness statement when equality holds in the integral condition.

Significance. If the main theorem were correct, the paper would provide a meaningful extension of the Orlicz-Minkowski problem to capillary convex bodies and would unify several recent capillary Lp results. The geometric inequalities in Section 3 appear to follow from known tools and are a plausible contribution: the Orlicz-Minkowski inequality (3.3) and Orlicz-Brunn-Minkowski inequality (3.4) are derived cleanly from the Alexandrov-Fenchel inequality of [48] and Jensen's inequality. However, the existence proof has a missing base point in the continuity argument, a mismatch between the normalized estimates and the unnormalized continuity path, and an incomplete Fredholm/surjectivity step. These are load-bearing defects, so the central existence theorem is not established in the present manuscript.

major comments (3)
  1. [Section 5, definition of H and proof of Theorem 1.1] The starting solution h=ℓ is not an element of the set H used in the continuity method. For h=ℓ we have det(ℓij+ℓδij)=1 and ℓ/h=1, so the coefficient in the defining condition of H reduces to the constant a=φ'(1)/φ(1)−n−1. If a≠0, the premise in H's definition is exactly ∫v=0, but the even function v=ℓ−c with c=(∫ℓ)/(∫1) has zero mean while ∫ℓv=∫ℓ²−c∫ℓ=Var(ℓ)>0 because ℓ is nonconstant. If a=0, the premise holds for every v, in particular v=ℓ, and ∫ℓ²>0. Thus ℓ∉H in either branch. The statement 'It is easy to see that h=ℓ is a solution to (5.1) for t=0' only checks the PDE, not membership in H. No alternative solution at t=0 is constructed. Consequently I is not shown to contain t=0, and the continuity method has no proven base point.
  2. [Section 4 versus Section 5] The a priori estimates of Section 4 are proved for the normalized equation (4.1), which contains the factor 1/|bΣ|; for example, Lemma 4.1 uses this factor to obtain the bound ∫ φ(ℓ/h) f ≤ n+1. The continuity equation (5.1) is not normalized. A solution h of (5.1) does not satisfy (4.1) with the same right-hand side f_t, and no rescaling argument is given. For general φ∈O, replacing h by |bΣ|^{-1/(n+1)}h does not preserve the term φ(ℓ/h) because φ is not homogeneous. The proof of closedness of I invokes the estimates (4.27), but those estimates do not apply to solutions of (5.1) as written. This is a second load-bearing gap in the existence proof.
  3. [Section 5, surjectivity of Lh] The proof that Lh is surjective on H is incomplete. Lemma 5.2 shows only that every v∈H∩Ker(Lh) satisfies the integral condition ∫(ℓφ'(ℓ/h)/(hφ(ℓ/h))−n−1)v det(hij+hδij)=0. The displayed chain in the proof of Theorem 1.1 then identifies (Ker(Lh))⊥ with the orthogonal complement of the set of v satisfying that integral condition. This identification requires the reverse inclusion, which is not proved and is not evident. Moreover, H itself is defined using h in the gauge condition, so H is not a linear subspace of C^{4,α}; arguments using 'Range(Lh)=H' and orthogonal complements in H therefore lack a clear functional-analytic setting. The implicit function theorem step is not justified on the basis of the lemmas stated.
minor comments (3)
  1. [Definition 1.2 and Section 5] The orthogonality condition in Definition 1.2 is stated with respect to a function f and involves f/(hφ(ℓ/h)), while the set H in Section 5 is defined with det(hij+hδij) and no f. The equivalence between these two formulations for solutions of (5.1) should be stated explicitly, especially because f_t changes along the continuity path.
  2. [Section 3, proof of Theorem 3.3] The proof of Theorem 3.3 cites 'Proposition 3.3', but no Proposition 3.3 appears in the paper; the reference should be to the appropriate definition, lemma, or variational formula.
  3. [Throughout] There are minor typographical issues, including 'theroy' in the Section 3 heading, a duplicated reference number [62] in the introduction, and the abstract phrase 'which can change our solutions to a spherical cap', which is unclear and should be rephrased.

Circularity Check

0 steps flagged · score 2.0 of 10

No circular reduction: the central derivation uses external tools; the missing continuity starting point is a soundness gap, not a circularity.

full rationale

The paper's main results are not obtained by assuming the theorem or by renaming fitted inputs. The Orlicz-Minkowski inequality (Theorem 3.2) is proved from Jensen's inequality plus the external Alexandrov-Fenchel inequality of [48], and the Orlicz-Brunn-Minkowski inequality (Theorem 3.3) is then derived from Definition 3.1 and Theorem 3.2. The a priori estimates in Section 4 are independent PDE estimates for the normalized equation (4.1). The only self-citation by the present authors is [60] (Zhu-Zhou-Xu), which appears in an introductory reference list on dual Orlicz-Brunn-Minkowski theory and is not load-bearing; this keeps the circularity score at 2 rather than 0. There is, however, a serious proof gap in Section 5 that is a correctness issue rather than a circular one. The proof states: 'It is easy to see that h = ℓ is a solution to (5.1) for t = 0,' but I is defined as the set of t for which (5.1) has a positive even solution in H. For h = ℓ, the gauge coefficient in the definition of H becomes the constant φ′(1)/φ(1) − n − 1. If this constant is nonzero, the premise in H's definition is exactly that v has zero mean, and v = ℓ − mean(ℓ) satisfies the premise but violates ∫ℓv = 0 because ℓ is nonconstant. If the constant is zero, the premise holds for every v, in particular v = ℓ, which again violates ∫ℓv = 0. Hence ℓ ∉ H, so the proof does not establish that 0 ∈ I and the continuity method has no proven starting point. Additionally, the Section 4 estimates apply to the normalized equation (4.1), whereas the continuation equation (5.1) is not normalized, so a volume-normalization argument is also missing. These are soundness defects in the proof, not reductions of the theorem to its own assumptions, and they do not raise the circularity score. The paper's derivation chain is otherwise self-contained with respect to external benchmarks. Nothing in the manuscript's own text admits a circular step: the ad hoc orthogonality condition is a nonstandard gauge, not a fitted parameter renamed as a prediction, and no uniqueness theorem is imported from the authors' own prior work to force the conclusion.

Assumptions & free parameters 0 free parameters · 6 assumptions · 2 invented entities

The central claim rests on the capillary framework from [44,48], the Aleksandrov-Fenchel inequality from [48], Orlicz addition from [15], and the class O assumptions. No numerical parameters are fitted. The orthogonality condition is an ad hoc normalization that the starting solution fails.

assumptions (6)
  • domain assumption Capillary support functions satisfy the Robin boundary condition ∇μh = cotθ h on ∂Cθ (from [44])
    Foundation of the capillary framework; quoted from [44, Proposition 2.4, 2.8].
  • domain assumption The Aleksandrov-Fenchel inequality for capillary convex hypersurfaces (Theorem 3.1, from [48])
    Used to prove Orlicz-Minkowski and Orlicz-Brunn-Minkowski inequalities; the paper cites [48] rather than proving it.
  • domain assumption Orlicz addition from Gardner-Hug-Weil [15] can be restricted to capillary bodies, preserving spherical convexity and Robin boundary condition (Lemma 3.1)
    The proof of Lemma 3.1 uses assumption A3 and relies on [15, 59] for spherical convexity.
  • standard math Regularity theory for Monge-Ampere equations with oblique boundary conditions (Lions-Trudinger-Urbas [34])
    Used in Lemma 4.5 for higher-order estimates.
  • standard math The spherical cap Cθ has support function ℓ with det(ℓij+ℓδij)=1 and volume |bCθ|
    Used as the explicit starting solution in the continuity method.
  • domain assumption The class O assumptions A1-A3 in (1.2) are imposed on φ
    These are hypotheses of Theorem 1.1; not derived.
invented entities (2)
  • capillary Orlicz surface area measure Sc_φ(bΣ,ω)
    purpose: Defines the prescribed measure in the capillary Orlicz-Minkowski problem.
    A new mathematical definition, not a physical postulate; no external falsifiable handle.
  • orthogonality condition (Definition 1.2)
    purpose: Removes kernel of the linearized operator in the continuity method.
    An ad hoc gauge condition that is not satisfied by the natural starting solution h=ℓ, breaking the proof.

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Pith. "Pith review of The capillary Orlicz-Minkowski problem." pith.science (2026). https://pith.science/paper/BQ2OJ66Y

@misc{pith2026250910859,
  author       = {Pith},
  title        = {Pith review of: The capillary Orlicz-Minkowski problem},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/BQ2OJ66Y}},
  note         = {Machine review of arXiv:2509.10859}
}
read the original abstract

In this paper, we introduce a Robin boundary analogue of the Orlicz-Minkowski problem, which seeks to find a capillary convex body with a prescribed capillary Orlicz surface area measure in the upper Euclidean half-space. We obtain the volume-normalized smooth solutions to the capillary even Orlicz-Minkowski problem by the continuity method. In addition, we also establish a capillary Orlicz-Brunn-Minkowski inequality and a capillary Orlicz-Minkowski inequality, which can change our solutions to a spherical cap under some conditions.

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Works this paper leans on

64 extracted references · 59 canonical work pages

  1. [48]

    X. Mei, G. Wang, L. Weng, C. Xia, Alexandrov-Fenchel inequalities for convex hypersurfaces in the half-space with capillary boundary II, Math. Z., 310 (2025) 17 pp

    work page 2025

  2. [1]

    Aleksandrov, On the theory of mixed volumes

    A.D. Aleksandrov, On the theory of mixed volumes. III. Extension of two the- orems of Minkowski on convex polyhedra to arbitrary convex bodies, Mat. Sbornik N.S., 3 (1938) 27–46

    work page 1938

  3. [2]

    Bianchi, K.J

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

    work page 2019

  4. [3]

    Bianchi, K.J

    G. Bianchi, K.J. B¨ or¨ oczky, A. Colesanti, Smoothness in the Lp Minkowski problem for p< 1, J. Geom. Anal., 30 (2020) 680–705

    work page 2020

  5. [4]

    B¨ or¨ oczky, E

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

    work page 2013

  6. [5]

    B¨ oro¨ oczky, E

    K.J. B¨ oro¨ oczky, E. Lutwak, D. Yang, G. Zhang, Y. Zhao, The Gauss image problem, Comm. Pure Appl. Math., 73 (2020) 1406–1452

    work page 2020

  7. [6]

    Bryan, M.N

    P. Bryan, M.N. Ivaki, J. Scheuer, A unified flow approach to smooth, even Lp-Minkowski problems, Anal. PDE, 12 (2019) 259–280

    work page 2019

  8. [7]

    Caffarelli, Interior W 2,p estimates for solutions of the Monge-Amp` ere equa- tion, Ann

    L.A. Caffarelli, Interior W 2,p estimates for solutions of the Monge-Amp` ere equa- tion, Ann. of Math., 131 (1990) 135–150

    work page 1990

Show all 64 references
  1. [8]

    Caffarelli, A localization property of viscosity solutions to the Monge- Amp` ere equation and their strict convexity, Ann

    L.A. Caffarelli, A localization property of viscosity solutions to the Monge- Amp` ere equation and their strict convexity, Ann. of Math., 131 (1990) 129–134

    1990

  2. [9]

    Chen, Q.-R

    S. Chen, Q.-R. Li, G. Zhu, On the Lp Monge-Amp` ere equation, J. Differential Equations, 263 (2017) 4997–5011

    2017

  3. [10]

    Cheng, S.-T

    S.-Y. Cheng, S.-T. Yau, On the regularity of the solution of the n-dimensional Minkowski problem, Comm. Pure Appl. Math., 29 (1976) 495–516

    1976

  4. [11]

    Chou, X.-J

    K.-S. Chou, X.-J. Wang, A variational theory of the Hessian equation, Comm. Pure Appl. Math., 54 (2001) 1029–1064. 24

    2001

  5. [12]

    Chou, X.-J

    K.-S. Chou, X.-J. Wang, The Lp-Minkowski problem and the Minkowski prob- lem in centroaffine geometry, Adv. Math., 205 (2006) 33–83

    2006

  6. [13]

    Fenchel, B

    W. Fenchel, B. Jessen, Mengenfunktionen und konvexe k¨ orper, Danske Vid. Selskab. Mat.-Fys. Medd., 16 (1938) 1–31

    1938

  7. [14]

    Firey, p-Means of convex bodies, Math

    W.J. Firey, p-Means of convex bodies, Math. Scand. 10 (1962) 17–24

    1962

  8. [15]

    Gardner, D

    R.J. Gardner, D. Hug, W. Weil, The Orlicz-Brunn-Minkowski theory: a general framework, additions, and inequalities, J. Differential Geom., 97 (2014) 427– 476

    2014

  9. [16]

    Gardner, D

    R.J. Gardner, D. Hug, W. Weil, D. Ye, The dual Orlicz-Brunn-Minkowski theory, J. Math. Anal. Appl., 430 (2015) 810–829

    2015

  10. [17]

    Gilbarg, N.S

    D. Gilbarg, N.S. Trudinger, Elliptic partial differential equations of second or- der, Springer, New York, 1983

    1983

  11. [18]

    P. Guan, C. Lin, On the equation det(uij +uδij) =upf on Sn, Preprint (1999)

    1999

  12. [19]

    Guang, Q.-R

    Q. Guang, Q.-R. Li, X.-J. Wang, TheLp-Minkowski problem with super-critical exponents, arXiv:2203.05099v1

    arXiv

  13. [20]

    Guang, Q.-R

    Q. Guang, Q.-R. Li, X.-J. Wang, Existence of convex hypersurfaces with pre- scribed centroaffine curvature, Trans. Amer. Math. Soc., 377 (2024) 841–862

    2024

  14. [21]

    Haberl, E

    C. Haberl, E. Lutwak, D. Yang, G. Zhang, The even Orlicz Minkowski problem, Adv. Math., 224 (2010), 2485–2510

    2010

  15. [22]

    He, Q.-R

    Y. He, Q.-R. Li, X.-J. Wang, Multiple solutions of the Lp-Minkowski problem, Calc. Var. Partial Differential Equations, 55 (2016) 13 pp

    2016

  16. [23]

    Y. Hu, M. N. Ivaki, Capillary curvature images, arXiv:2505.12921v1

    arXiv

  17. [24]

    Y. Hu, M. N. Ivaki, J. Scheuer, Capillary Christoffel-Minkowski problem, arXiv:2504.09320v1

    arXiv

  18. [25]

    Y. Hu, Y. Wei, B. Yang, T. Zhou, A complete family of Alexandrov-Fenchel inequalities for convex capillary hypersurfaces in the half-space, Math. Ann., 390 (2024) 3039–3075

    2024

  19. [26]

    Huang, Q

    Y. Huang, Q. Lu, On the regularity of the Lp Minkowski problem, Adv. in Appl. Math., 50 (2013) 268–280

    2013

  20. [27]

    Huang, E

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

    2016

  21. [28]

    Huang, E

    Y. Huang, E. Lutwak, D. Yang, G. Zhang, The Lp-Aleksandrov problem for Lp-integral curvature, J. Differential Geom., 110 (2018) 29pp

    2018

  22. [29]

    Huang, D

    Y. Huang, D. Yang, G. Zhang, Minkowski problems for geometric measures, Bull. Amer. Math. Soc., 62 (2025) 359–425

    2025

  23. [30]

    D. Hug, E. Lutwak, D. Yang, G. Zhang, On the Lp Minkowski problem for polytopes, Discrete Comput. Geom., 33 (2005) 699–715

    2005

  24. [31]

    H. Jian, J. Lu, Existence of solutions to the Orlicz-Minkowski problem, Adv. Math., 344 (2019) 262–288. 25

    2019

  25. [32]

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

    2015

  26. [33]

    Lewy, On differential geometric in the large

    H. Lewy, On differential geometric in the large. I. Minkowski’s problem, Trans. Amer. Math. Soc., 43 (1938) 258–270

    1938

  27. [34]

    Lions, N.S

    P.-L. Lions, N.S. Trudinger, J.I.E. Urbas, The Neumann problem for equations of Monge-Amp` ere type, Comm. Pure Appl. Math., 39 (1986) 539–563

    1986

  28. [35]

    Y. Liu, J. Lu, A flow method for the dual Orlicz-Minkowski problem, Trans. Amer. Math. Soc., 373 (2020) 5833–5853

    2020

  29. [36]

    Lu, X.-J

    J. Lu, X.-J. Wang, Rotationally symmetric solutions to theLp-Minkowski prob- lem, J. Differential Equations, 254 (2013) 983–1005

    2013

  30. [37]

    Lutwak, The Brunn-Minkowski-Firey Theory I: Mixed volumes and the Minkowski problem, J

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

    1993

  31. [38]

    Lutwak, V

    E. Lutwak, V. Oliker, On the regularity of solutions to a generalization of the Minkowski problem, J. Differential Geom., 41 (1995) 227–246

    1995

  32. [39]

    Lutwak, D

    E. Lutwak, D. Yang, G. Zhang, On the Lp-Minkowski problem, Trans. Amer. Math. Soc., 356 (2004) 4359–4370

    2004

  33. [40]

    Lutwak, D

    E. Lutwak, D. Yang, G. Zhang, Orlicz projection bodies, Adv. Math., 223 (2010) 220–242

    2010

  34. [41]

    Lutwak, D

    E. Lutwak, D. Yang, G. Zhang, Orlicz centroid bodies, J. Differential Geom., 84 (2010) 365–387

    2010

  35. [42]

    Lutwak, D

    E. Lutwak, D. Yang, G. Zhang, Lp dual curvature measures, Adv. Math., 329 (2018) 85–132

    2018

  36. [43]

    X. Mei, G. Wang, L. Weng, A constrained mean curvature flow and Alexandrov- Fenchel inequalities, Int. Math. Res. Not. IMRN 1, (2024) 152–174

    2024

  37. [44]

    X. Mei, G. Wang, L. Weng, The capillary Minkowski problem, Adv. Math., 469 (2025) 29 pp

    2025

  38. [45]

    X. Mei, G. Wang, L. Weng, Convex capillary hypersurfaces of prescribed cur- vature problem, arXiv:2504.14392v1

    arXiv

  39. [46]

    X. Mei, G. Wang, L. Weng, The capillary Lp-Minkowski problem, arXiv:2505.07746v2

    arXiv

  40. [47]

    X. Mei, G. Wang, L. Weng, The capillary Gauss curvature flow, arXiv:2506.09840v1

    arXiv

  41. [49]

    X. Mei, L. Weng, A fully nonlinear locally constrained curvature flow for cap- illary hypersurface, Ann. Global Anal. Geom., 67 (2025) 17 pp

    2025

  42. [50]

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

    H. Minkowski, Allgemeine Lehrs¨ atze ¨ uber die konvexen Polyeder, Nachr. Ges. Wiss. G¨ ott., (1897) 198–219. 26

  43. [51]

    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 (1953) 337–394

    1953

  44. [52]

    Pogorelov, The Minkowski multidimensional problem, V.H

    A.V. Pogorelov, The Minkowski multidimensional problem, V.H. Winston, Washington, D.C., 1978

    1978

  45. [53]

    Schneider, Convex bodies: the Brunn-Minkowski theory

    R. Schneider, Convex bodies: the Brunn-Minkowski theory. 2nd edn., Ency- clopedia of Mathematics and Its Applications, vol. 151, Cambridge University Press, Cambridge, (2014)

    2014

  46. [54]

    Shenfeld, R

    Y. Shenfeld, R. van Handel, Mixed volumes and the Bochner method, Proc. Amer. Math. Soc., 147 (2019) 5385–5402

    2019

  47. [55]

    Shenfeld, R

    Y. Shenfeld, R. van Handel, The extremals of Minkowski’s quadratic inequality, Duke Math. J., 171 (2022) 957–1027

    2022

  48. [56]

    Shenfeld, R

    Y. Shenfeld, R. van Handel, The extremals of the Alexandrov-Fenchel inequality for convex polytopes, Acta Math., 231 (2023) 89–204

    2023

  49. [57]

    G. Wang, L. Weng, C. Xia, Alexandrov-Fenchel inequalities for convex hy- persurfaces in the half-space with capillary boundary, Math. Ann., 388 (2024) 2121–2154

    2024

  50. [58]

    G. Wang, L. Weng, C. Xia, A Minkowski-type inequality for capillary hyper- surfaces in a half-space, J. Funct. Anal., 287 (2024) 22 pp

    2024

  51. [59]

    D. Xi, H. Jin, G. Leng, The Orlicz Brunn-Minkowski inequality, Adv. Math., 260 (2014) 350–374

    2014

  52. [60]

    B. Zhu, J. Zhou, W. Xu, Dual Orlicz-Brunn-Minkowski theory, Adv. Math., 264 (2014) 700–725

    2014

  53. [61]

    Zhu, The logarithmic Minkowski problem for polytopes, Adv

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

    2014

  54. [62]

    Zhu, The Lp Minkowski problem for polytopes for 0 < p <1, J

    G. Zhu, The Lp Minkowski problem for polytopes for 0 < p <1, J. Funct. Anal., 269 (2015) 1070–1094

    2015

  55. [63]

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

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

    2015

  56. [64]

    D. Zou, G. Xiong, Orlicz-John ellipsoids, Adv. Math., 265 (2014) 132–168. Xudong Wang, xdwang@snnu.edu.cn School of Mathematics and Statistics, Shaanxi Normal University, Xi’an, 710119, China Baocheng Zhu, bczhu@snnu.edu.cn School of Mathematics and Statistics, Shaanxi Normal ...

    2014

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