Explicit solutions to Christoffel-Minkowski problems and Hessian equations under rotational symmetries

Fabian Mussnig; Jacopo Ulivelli

arxiv: 2508.11600 · v2 · pith:2E2B2KEYnew · submitted 2025-08-15 · 🧮 math.MG · math.AP· math.DG· math.FA

Explicit solutions to Christoffel-Minkowski problems and Hessian equations under rotational symmetries

Fabian Mussnig , Jacopo Ulivelli This is my paper

Pith reviewed 2026-05-21 22:22 UTC · model grok-4.3

classification 🧮 math.MG math.APmath.DGmath.FA

keywords Christoffel-Minkowski problemconvex bodiesrotational symmetryMonge-Ampère equationsHessian equationssupport functionarea measures

0 comments

The pith

The Christoffel-Minkowski problem for convex bodies of revolution admits an explicit solution via a formula for the support function based on first moments of the measure.

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

This paper solves the Christoffel-Minkowski problem explicitly when the convex body and the data are rotationally symmetric. The symmetry reduces the governing mixed Monge-Ampère equation to an integrable ordinary differential equation. As a result, the support function of the convex body is recovered by a direct formula involving integrals of the prescribed measure over spherical caps. The same reduction provides explicit convex solutions for mixed area measures and for the Dirichlet problem of k-Hessian equations under radial symmetry.

Core claim

Under the assumption of rotational symmetry, an explicit representation formula is derived for the support function of a convex body of revolution that solves the Christoffel-Minkowski problem for a given measure. The conditions required on the measure are formulated in terms of its first moments computed over spherical caps. The method extends to constructing explicit convex solutions of mixed Monge-Ampère equations on Euclidean space when restricted to radial symmetry, with data specified on open balls, and includes the Dirichlet problem for k-Hessian equations as a special case.

What carries the argument

Reduction of mixed Monge-Ampère equations to radially symmetric form allowing explicit integration of the support function from the measure data.

If this is right

The explicit formula directly yields the convex body from any measure satisfying the first-moment conditions on caps.
Existence for mixed area measures follows from the same radial integration procedure.
The k-Hessian Dirichlet problem on R^n possesses explicit radial solutions when symmetry is imposed.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

These formulas could benchmark general numerical solvers for the non-symmetric problem.
The radial reduction technique may apply to other fully nonlinear equations with symmetry assumptions.
One could test whether small perturbations from rotational symmetry preserve approximate explicit solutions.

Load-bearing premise

Both the convex body and the measure must be rotationally symmetric about the same axis for the reduction to ordinary differential equations to hold.

What would settle it

Construct a rotationally symmetric measure meeting the moment conditions and check whether the body defined by the explicit support function has the original measure as its mixed area measure.

read the original abstract

An explicit solution to the Christoffel-Minkowski problem for convex bodies of revolution is presented. The conditions on the prescribed measure involve only first moments over spherical caps, and the support function of the resulting convex body is given by an explicit representation formula in terms of the measure. More generally, existence problems for mixed area measures are addressed. The approach relies on constructing explicit convex solutions to mixed Monge-Amp\`ere equations on $\mathbb{R}^n$ under the assumption of radial symmetry, with the conditions on the measure being expressed through its values on open balls. As a special case, the Dirichlet problem for $k$-Hessian equations on $\mathbb{R}^n$ is treated.

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.

Desk Editor's Note private letter to a colleague

The paper gives explicit formulas for the support function of rotationally symmetric convex bodies that solve the Christoffel-Minkowski problem, built from first moments over spherical caps.

read the letter

This paper gives explicit formulas for the support function of convex bodies of revolution that solve the Christoffel-Minkowski problem. The support function is written directly in terms of the first moments of the measure over spherical caps. They reduce the mixed Monge-Ampère equation to a radial problem by assuming rotational symmetry and integrate it to get a concrete expression. The same method covers mixed area measures more generally and the Dirichlet problem for k-Hessian equations on R^n under radial symmetry. This turns an existence question into something you can write down for the symmetric case. The moment conditions are natural and fit the radial setting without extra assumptions. The construction follows from integrating the reduced operator, and there is no sign of circularity or fitted parameters. The main limitation is the rotational symmetry requirement. That is what makes the explicit integration possible, but it keeps the result from applying to general convex bodies. The paper states this scope clearly. The symmetry reduction itself is a standard step that appears valid here. A reader working in convex geometry or fully nonlinear PDEs would find the formulas useful for examples or checks in the symmetric setting. Someone outside that subfield would not get much from it. I would send this to peer review. The explicit formulas are concrete enough to merit checking the derivations in detail.

Referee Report

1 major / 2 minor

Summary. The manuscript presents explicit solutions to the Christoffel-Minkowski problem for convex bodies of revolution. Under the assumption of rotational symmetry, the mixed Monge-Ampère equations reduce to a one-dimensional radial setting, yielding an explicit integral representation for the support function expressed solely in terms of first-moment conditions on the prescribed measure over spherical caps (equivalently, values on open balls for the radial measure). The paper extends the method to existence questions for mixed area measures and treats the Dirichlet problem for k-Hessian equations on R^n as a special case.

Significance. If the explicit constructions hold, the work supplies concrete, closed-form solutions in a setting where general existence theorems exist but explicit formulas are uncommon. The symmetry reduction converts the mixed Hessian operator into an integrable radial form whose output is determined directly by the moment data, providing a parameter-free derivation that could serve as a benchmark for numerical methods and further analysis of Hessian-type equations in convex geometry.

major comments (1)

The central claim that the constructed support function satisfies the prescribed mixed area measure rests on the radial integration step; a direct verification that the resulting function is convex and reproduces the given first moments (without additional regularity assumptions on the measure) would strengthen the argument. This appears in the derivation following the symmetry reduction but is not fully detailed in the provided outline.

minor comments (2)

Clarify the precise relation between the spherical-cap moments and the open-ball values of the radial measure in the statement of the main theorem to avoid ambiguity for readers.
The abstract mentions 'more generally, existence problems for mixed area measures'; a brief comparison with existing non-explicit existence results would help situate the contribution.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and constructive suggestion. We address the major comment below and will revise the manuscript to incorporate a more detailed verification.

read point-by-point responses

Referee: The central claim that the constructed support function satisfies the prescribed mixed area measure rests on the radial integration step; a direct verification that the resulting function is convex and reproduces the given first moments (without additional regularity assumptions on the measure) would strengthen the argument. This appears in the derivation following the symmetry reduction but is not fully detailed in the provided outline.

Authors: We agree that expanding the verification would improve clarity. In the manuscript the support function is obtained from the explicit integral formula arising from the radial reduction; convexity follows because the integrand consists of support functions of convex sets (spherical caps) and the integral preserves convexity. Reproduction of the first moments holds by direct application of Fubini’s theorem to the radial integrals, which is valid for arbitrary positive Radon measures on the sphere without extra regularity. In the revision we will add a dedicated paragraph immediately after the symmetry reduction that carries out this verification step by step, including the passage to general measures via weak approximation. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper reduces the Christoffel-Minkowski problem and mixed Monge-Ampère equations to a radially symmetric one-dimensional setting under the assumption of rotational symmetry for convex bodies of revolution. This permits explicit integration yielding a support function expressed directly in terms of first-moment conditions on spherical caps (or values on open balls for the radial measure). The derivation chain consists of standard symmetry reductions and direct integration steps that follow from the form of the radial mixed Hessian operator; no step equates a claimed prediction or result to its own inputs by construction, and no load-bearing self-citation or uniqueness theorem imported from prior author work is invoked. The explicit representation formula is obtained from the symmetry assumptions without circular dependencies, rendering the central claims self-contained against the stated hypotheses.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the domain assumption of rotational symmetry together with standard convexity and measure-theoretic background; no free parameters or new entities are introduced in the abstract.

axioms (2)

domain assumption Convex bodies and measures are rotationally symmetric (of revolution).
This symmetry is invoked to reduce the mixed Monge-Ampère equations to a radially symmetric setting that permits explicit integration.
domain assumption The prescribed measure satisfies first-moment conditions over spherical caps.
These integral conditions replace pointwise requirements and are stated as sufficient for the explicit formula to work.

pith-pipeline@v0.9.0 · 5653 in / 1280 out tokens · 63689 ms · 2026-05-21T22:22:36.610958+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/AlexanderDuality alexander_duality_circle_linking unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Radial symmetry reduces MA(u1,...,un; D_r) = κ_n ∏ p_{u_i}(r) (Lemma 3.1); monotonicity forces convexity

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

39 extracted references · 39 canonical work pages

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L. C AFFARELLI , L. N IRENBERG , AND J. S PRUCK , The Dirichlet problem for nonlinear second-order elliptic equations. III. Functions of the eigenvalues of the Hessian. Acta Math. 155 (1985), 261–301. ↑ 4

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K.-S. C HOU AND X.-J. W ANG, Minkowski problems for complete noncompact convex hypersurfaces. Topol. Methods Nonlinear Anal. 6 (1995), 151–162. ↑ 5

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E. B. C HRISTOFFEL , Ueber die Bestimmung der Gestalt einer krummen Oberfl ¨ache durch lokale Messungen auf derselben. J. Reine Angew. Math. 64 (1865), 193–209. ↑ 1

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A. C OLESANTI AND D. H UG, Hessian measures of convex functions and applications to area measures . J. Lond. Math. Soc. (2) 71 (2005), 221–235. ↑ 6

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C OLESANTI , M

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C OLESANTI , M

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C OLESANTI AND P

A. C OLESANTI AND P. SALANI , Hessian equations in non-smooth domains. Nonlinear Anal. 38 (1999), 803–

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F IGALLI , The Monge-Amp `ere Equation and Its Applications

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W. J. F IREY , Christoffel’s problem for general convex bodies. Mathematika 15 (1968), 7–21. ↑ 1

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W. J. F IREY , Intermediate Christoffel-Minkowski problems for figures of revolution. Israel J. Math. 8 (1970), 384–390. ↑ 1, 9, 22

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G UAN, C

P. G UAN, C. L IN, AND X.-N. M A, The Christoffel-Minkowski problem. II. Weingarten curvature equations. Chinese Ann. Math. Ser. B 27 (2006), 595–614. ↑ 1 23

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G UAN AND X.-N

P. G UAN AND X.-N. M A, The Christoffel-Minkowski problem. I. Convexity of solutions of a Hessian equation. Invent. Math. 151 (2003), 553–577. ↑ 1

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G UAN, X.-N

P. G UAN, X.-N. M A, AND F. Z HOU, The Christofel-Minkowski problem. III. Existence and convexity of admissible solutions. Comm. Pure Appl. Math. 59 (2006), 1352–1376. ↑ 1

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H UANG , D

Y. H UANG , D. YANG , AND G. Z HANG , Minkowski problems for geometric measures. Bull. Amer. Math. Soc. (N.S.) 62 (2025), 359–425. ↑ 2

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D. H UG, F. MUSSNIG , AND J. U LIVELLI , Additive kinematic formulas for convex functions. Canad. J. Math., to appear. ↑ 7, 11, 22

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J. K NOERR AND J. U LIVELLI , From valuations on convex bodies to convex functions. Math. Ann. 390 (2024), 5987–6011. ↑ 14

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L IU, X.-N

P. L IU, X.-N. M A, AND L. X U, A Brunn-Minkowski inequality for the Hessian eigenvalue in three- dimensional convex domain. Adv. Math. 225 (2010), 1616–1633. ↑ 4

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M A AND L

X.-N. M A AND L. X U, The convexity of solution of a class Hessian equation in bounded convex domain in R3. J. Funct. Anal. 255 (2008), 1713–1723. ↑ 4

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M. A. M OUAMINE AND F. M USSNIG , The vectorial Hadwiger theorem on convex functions . Preprint, arXiv:2504.04952 (2025). ↑ 7

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R. T. R OCKAFELLAR , Convex Analysis. Princeton Landmarks in Mathematics, Princeton University Press, Princeton, NJ, 1997. ↑ 5, 6, 10, 11

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[31]

S ALANI , Boundary blow-up problems for Hessian equations

P. S ALANI , Boundary blow-up problems for Hessian equations. Manuscripta Math.96 (1998), 281–294. ↑ 4, 9

work page 1998
[32]

S ALANI , Convexity of solutions and Brunn-Minkowski inequalities for Hessian equations inR3

P. S ALANI , Convexity of solutions and Brunn-Minkowski inequalities for Hessian equations inR3. Adv. Math. 229 (2012), 1924–1948. ↑ 4

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S CHNEIDER , Convex Bodies: The Brunn–Minkowski Theory

R. S CHNEIDER , Convex Bodies: The Brunn–Minkowski Theory. Second expanded ed., Encyclopedia of Math- ematics and its Applications, vol. 151, Cambridge University Press, Cambridge, 2014. ↑ 1, 2, 5, 16, 22

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N. S. T RUDINGER , On the Dirichlet problem for Hessian equations. Acta Math. 175 (1995), 151–164. ↑ 4, 6

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N. S. T RUDINGER , Weak solutions of Hessian equations . Comm. Partial Differential Equations 22 (1997), 1251–1261. ↑ 4

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[36]

N. S. T RUDINGER AND X.-J. W ANG, Hessian measures. I. Topol. Methods Nonlinear Anal. 10 (1997), 225–

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[37]

N. S. T RUDINGER AND X.-J. W ANG, Hessian measures. II. Ann. of Math. (2) 150 (1999), 579–604. ↑ 4, 6

work page 1999
[38]

N. S. T RUDINGER AND X.-J. W ANG, Hessian measures. III. J. Funct. Anal. 193 (2002), 1–23. ↑ 4, 7

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U LIVELLI , Entire Monge–Amp`ere equations and weighted Minkowski problems

J. U LIVELLI , Entire Monge–Amp`ere equations and weighted Minkowski problems. Commun. Contemp. Math., to appear. ↑ 5 Fabian Mussnig Institut f¨ur Diskrete Mathematik und Geometrie TU Wien Wiedner Hauptstraße 8-10/1046 1040 Wien, Austria e-mail: fabian.mussnig@tuwien.ac.at Jacopo Ulivelli Institut f¨ur Diskrete Mathematik und Geometrie TU Wien Wiedner Hau...

work page

[1] [1]

A. D. A LEKSANDROV , Dirichlet’s problem for the equation Det∥zi j∥ = ϕ(z1, . . .,zn,z,x1, . . .,xn). I (Russian). Vestnik Leningrad. Univ. Ser. Mat. Meh. Astr. 13 (1958), 5–24. English translation in: A. D. Aleksandrov, Selected Works Part I: Selected Scientific Papers, Classic of Soviet Mathematics, 4, Gordon and Branch Pub- lishers, Amsterdam, 1996. ↑ 6

work page 1958

[2] [2]

I. J. B AKELMAN , Convex Analysis and Nonlinear Geometric Elliptic Equations . Springer-Verlag, Berlin,

work page

[3] [3]

B ERG, Corps convexes et potentiels sph´eriques

C. B ERG, Corps convexes et potentiels sph´eriques. Mat.-Fys. Medd. Danske Vid. Selsk. 37 (1969), 64pp. ↑ 1

work page 1969

[4] [4]

K. J. B ¨OR ¨OCZKY , A. F IGALLI , AND J. P. RAMOS , The Isoperimetric Inequality, the Brunn–Minkowski The- ory and Minkowski Type Monge–Amp `ere Equations on the Sphere . European Mathematical Society (EMS), Z¨urich, to appear. ↑ 2

work page

[5] [5]

B RAUNER , G

L. B RAUNER , G. C. H OFST ¨ATTER , AND O. O RTEGA -MORENO , Lefschetz operators on convex valuations. Preprint, arXiv:2402.14731 (2024). ↑ 3

work page arXiv 2024

[6] [6]

B RAUNER , G

L. B RAUNER , G. C. H OFST ¨ATTER , AND O. O RTEGA -M ORENO , The Klain approach to zonal valuations . Preprint, arXiv:2410.18651 (2024). ↑ 22

work page arXiv 2024

[7] [7]

B RAUNER , G

L. B RAUNER , G. C. H OFST ¨ATTER , AND O. O RTEGA -M ORENO , Mixed Christoffel-Minkowski problems for bodies of revolution. Preprint, arXiv:2508.09794 (2025). ↑ 1, 2, 3, 16, 20, 21, 22

work page arXiv 2025

[8] [8]

B RAUNER , G

L. B RAUNER , G. C. H OFST ¨ATTER , AND O. O RTEGA -MORENO , The Christoffel problem for the disk area measure. Preprint, arXiv:2508.09800 (2025). ↑ 21

work page arXiv 2025

[9] [9]

B RYAN, M

P. B RYAN, M. N. I VAKI, AND J. SCHEUER , Christoffel-Minkowski flows. Trans. Amer. Math. Soc.376 (2023), 2373–2393. ↑ 1

work page 2023

[10] [10]

C AFFARELLI , L

L. C AFFARELLI , L. N IRENBERG , AND J. S PRUCK , The Dirichlet problem for nonlinear second-order elliptic equations. III. Functions of the eigenvalues of the Hessian. Acta Math. 155 (1985), 261–301. ↑ 4

work page 1985

[11] [11]

C HOU AND X.-J

K.-S. C HOU AND X.-J. W ANG, Minkowski problems for complete noncompact convex hypersurfaces. Topol. Methods Nonlinear Anal. 6 (1995), 151–162. ↑ 5

work page 1995

[12] [12]

E. B. C HRISTOFFEL , Ueber die Bestimmung der Gestalt einer krummen Oberfl ¨ache durch lokale Messungen auf derselben. J. Reine Angew. Math. 64 (1865), 193–209. ↑ 1

work page

[13] [13]

C OLESANTI AND D

A. C OLESANTI AND D. H UG, Hessian measures of convex functions and applications to area measures . J. Lond. Math. Soc. (2) 71 (2005), 221–235. ↑ 6

work page 2005

[14] [14]

C OLESANTI , M

A. C OLESANTI , M. L UDWIG , AND F. M USSNIG , Hessian valuations . Indiana Univ. Math. J. 69 (2020), 1275–1215. ↑ 6, 7, 9

work page 2020

[15] [15]

C OLESANTI , M

A. C OLESANTI , M. L UDWIG , AND F. M USSNIG , The Hadwiger theorem on convex functions, III: Steiner formulas and mixed Monge-Amp `ere measures. Calc. Var. Partial Differential Equations 61 (2022), Art. 181. ↑ 7

work page 2022

[16] [16]

C OLESANTI AND P

A. C OLESANTI AND P. SALANI , Hessian equations in non-smooth domains. Nonlinear Anal. 38 (1999), 803–

work page 1999

[17] [17]

F IGALLI , The Monge-Amp `ere Equation and Its Applications

A. F IGALLI , The Monge-Amp `ere Equation and Its Applications . European Mathematical Society (EMS), Z¨urich, 2017. ↑ 5, 6, 12

work page 2017

[18] [18]

W. J. F IREY , Christoffel’s problem for general convex bodies. Mathematika 15 (1968), 7–21. ↑ 1

work page 1968

[19] [19]

W. J. F IREY , Intermediate Christoffel-Minkowski problems for figures of revolution. Israel J. Math. 8 (1970), 384–390. ↑ 1, 9, 22

work page 1970

[20] [20]

G UAN, C

P. G UAN, C. L IN, AND X.-N. M A, The Christoffel-Minkowski problem. II. Weingarten curvature equations. Chinese Ann. Math. Ser. B 27 (2006), 595–614. ↑ 1 23

work page 2006

[21] [21]

G UAN AND X.-N

P. G UAN AND X.-N. M A, The Christoffel-Minkowski problem. I. Convexity of solutions of a Hessian equation. Invent. Math. 151 (2003), 553–577. ↑ 1

work page 2003

[22] [22]

G UAN, X.-N

P. G UAN, X.-N. M A, AND F. Z HOU, The Christofel-Minkowski problem. III. Existence and convexity of admissible solutions. Comm. Pure Appl. Math. 59 (2006), 1352–1376. ↑ 1

work page 2006

[23] [23]

H UANG , D

Y. H UANG , D. YANG , AND G. Z HANG , Minkowski problems for geometric measures. Bull. Amer. Math. Soc. (N.S.) 62 (2025), 359–425. ↑ 2

work page 2025

[24] [24]

D. H UG, F. M USSNIG , AND J. U LIVELLI , Kubota-type formulas and supports of mixed measures . Preprint, arXiv:2401.16371 (2024). ↑ 7, 8, 9, 14

work page arXiv 2024

[25] [25]

D. H UG, F. MUSSNIG , AND J. U LIVELLI , Additive kinematic formulas for convex functions. Canad. J. Math., to appear. ↑ 7, 11, 22

work page

[26] [26]

K NOERR AND J

J. K NOERR AND J. U LIVELLI , From valuations on convex bodies to convex functions. Math. Ann. 390 (2024), 5987–6011. ↑ 14

work page 2024

[27] [27]

L IU, X.-N

P. L IU, X.-N. M A, AND L. X U, A Brunn-Minkowski inequality for the Hessian eigenvalue in three- dimensional convex domain. Adv. Math. 225 (2010), 1616–1633. ↑ 4

work page 2010

[28] [28]

M A AND L

X.-N. M A AND L. X U, The convexity of solution of a class Hessian equation in bounded convex domain in R3. J. Funct. Anal. 255 (2008), 1713–1723. ↑ 4

work page 2008

[29] [29]

M. A. M OUAMINE AND F. M USSNIG , The vectorial Hadwiger theorem on convex functions . Preprint, arXiv:2504.04952 (2025). ↑ 7

work page arXiv 2025

[30] [30]

R. T. R OCKAFELLAR , Convex Analysis. Princeton Landmarks in Mathematics, Princeton University Press, Princeton, NJ, 1997. ↑ 5, 6, 10, 11

work page 1997

[31] [31]

S ALANI , Boundary blow-up problems for Hessian equations

P. S ALANI , Boundary blow-up problems for Hessian equations. Manuscripta Math.96 (1998), 281–294. ↑ 4, 9

work page 1998

[32] [32]

S ALANI , Convexity of solutions and Brunn-Minkowski inequalities for Hessian equations inR3

P. S ALANI , Convexity of solutions and Brunn-Minkowski inequalities for Hessian equations inR3. Adv. Math. 229 (2012), 1924–1948. ↑ 4

work page 2012

[33] [33]

S CHNEIDER , Convex Bodies: The Brunn–Minkowski Theory

R. S CHNEIDER , Convex Bodies: The Brunn–Minkowski Theory. Second expanded ed., Encyclopedia of Math- ematics and its Applications, vol. 151, Cambridge University Press, Cambridge, 2014. ↑ 1, 2, 5, 16, 22

work page 2014

[34] [34]

N. S. T RUDINGER , On the Dirichlet problem for Hessian equations. Acta Math. 175 (1995), 151–164. ↑ 4, 6

work page 1995

[35] [35]

N. S. T RUDINGER , Weak solutions of Hessian equations . Comm. Partial Differential Equations 22 (1997), 1251–1261. ↑ 4

work page 1997

[36] [36]

N. S. T RUDINGER AND X.-J. W ANG, Hessian measures. I. Topol. Methods Nonlinear Anal. 10 (1997), 225–

work page 1997

[37] [37]

N. S. T RUDINGER AND X.-J. W ANG, Hessian measures. II. Ann. of Math. (2) 150 (1999), 579–604. ↑ 4, 6

work page 1999

[38] [38]

N. S. T RUDINGER AND X.-J. W ANG, Hessian measures. III. J. Funct. Anal. 193 (2002), 1–23. ↑ 4, 7

work page 2002

[39] [39]

U LIVELLI , Entire Monge–Amp`ere equations and weighted Minkowski problems

J. U LIVELLI , Entire Monge–Amp`ere equations and weighted Minkowski problems. Commun. Contemp. Math., to appear. ↑ 5 Fabian Mussnig Institut f¨ur Diskrete Mathematik und Geometrie TU Wien Wiedner Hauptstraße 8-10/1046 1040 Wien, Austria e-mail: fabian.mussnig@tuwien.ac.at Jacopo Ulivelli Institut f¨ur Diskrete Mathematik und Geometrie TU Wien Wiedner Hau...

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