The Orlicz-Gauss image problem for pseudo-cones and its associated spherical optimal transport

Siqi Lei; Xudong Wang

arxiv: 2601.08170 · v2 · pith:CW4AHQHMnew · submitted 2026-01-13 · 🧮 math.FA

The Orlicz-Gauss image problem for pseudo-cones and its associated spherical optimal transport

Siqi Lei , Xudong Wang This is my paper

Pith reviewed 2026-05-21 16:49 UTC · model grok-4.3

classification 🧮 math.FA

keywords Orlicz-Gauss image problempseudo-conesspherical optimal transportvariational methodrestrictive techniqueconvex geometryexistence conditions

0 comments

The pith

A necessary and sufficient condition determines when the Orlicz-Gauss image problem has solutions for pseudo-cones

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

Pseudo-cones serve as noncompact counterparts to convex bodies in convex geometry. The paper derives a necessary and sufficient condition for the existence of solutions to the Orlicz-Gauss image problem in this setting. It further connects the problem to spherical optimal transport. The proof combines the variational method with a novel restrictive technique. This strengthens Schneider's earlier result by improving the constant factor.

Core claim

The paper establishes a necessary and sufficient condition for the existence of solutions to the Orlicz-Gauss image problem for pseudo-cones. The condition is obtained by merging the variational method with a novel restrictive technique. This approach strengthens the original result of Schneider up to a constant factor and demonstrates a connection to spherical optimal transport.

What carries the argument

The Orlicz-Gauss image problem for pseudo-cones, solved using a variational method paired with a novel restrictive technique to produce a necessary and sufficient existence condition while linking the problem to spherical optimal transport.

If this is right

A solution to the Orlicz-Gauss image problem for a given pseudo-cone exists exactly when the derived condition holds.
The Orlicz-Gauss image problem for pseudo-cones is equivalent to a corresponding spherical optimal transport problem.
The restrictive technique yields an improved constant in the existence result relative to prior work on the problem.

Where Pith is reading between the lines

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

The restrictive technique may apply to existence questions for other image problems involving unbounded convex sets.
The link to spherical optimal transport suggests possible numerical schemes for computing solutions in the pseudo-cone case.
The result could extend to related problems in asymptotic convex geometry where noncompactness plays a central role.

Load-bearing premise

The novel restrictive technique combined with the variational method suffices to produce both the necessary-and-sufficient characterization and the constant improvement in the noncompact pseudo-cone setting.

What would settle it

Construct a specific pseudo-cone satisfying the stated condition for which no solution to the Orlicz-Gauss image problem exists, or find a pseudo-cone with a solution despite failing the condition.

read the original abstract

Pseudo-cones serve as the noncompact counterpart of convex bodies in convex geometry. This paper establishes a necessary and sufficient condition for the existence of solutions to the Orlicz-Gauss image problem for pseudo-cones and further demonstrates its connection to spherical optimal transport. Our approach combines the variational method with a novel restrictive technique, thereby strengthening the original result of Schneider up to a constant factor.

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 claims a necessary-and-sufficient condition for the Orlicz-Gauss image problem on pseudo-cones plus a spherical optimal transport link and a constant improvement over Schneider, but the reversibility of the restrictive technique in the noncompact case is the spot that needs checking.

read the letter

The main takeaway is that the authors give a necessary and sufficient existence condition for the Orlicz-Gauss image problem on pseudo-cones, connect it to spherical optimal transport, and improve Schneider's constant by combining the variational method with a new restrictive technique. This is a direct extension of the compact-body case to the noncompact setting that pseudo-cones are meant to handle. The transport reformulation is a clean way to recast the problem and could make future work easier. If the proofs are complete, the constant-factor gain is a concrete, usable improvement over the earlier result. The approach is explicit enough that a reader can see where the new ideas sit relative to prior work. The soft spot sits in the necessity direction. The restrictive technique narrows the admissible class before minimization, and in the noncompact setting pseudo-cones can have nontrivial recession cones. For necessity to hold without extra assumptions at infinity, every solution of the original problem must lift from a solution of the restricted problem. The abstract states the condition as necessary and sufficient, but it does not show the reversibility step in detail. If that argument relies on implicit decay or growth conditions, the necessity claim would be weaker than advertised. This is the precise point the stress-test note flags, and it is worth a close look in the proofs rather than a minor technicality. The paper is written for people already working in convex geometry and Brunn-Minkowski theory who know the Gauss image problem. A reader who has read Schneider will see the incremental advance and the transport connection as useful additions. It is too specialized to draw a wider audience. I would bring it to a reading group to walk through the restrictive technique and the Euler-Lagrange derivation. I would not cite it in my own work until the necessity argument is verified. It deserves peer review because the claim is stated sharply, the setting is natural, and the method is described clearly enough for referees to evaluate the central step.

Referee Report

2 major / 2 minor

Summary. The paper claims to establish a necessary and sufficient condition for the existence of solutions to the Orlicz-Gauss image problem for pseudo-cones (the noncompact analogue of convex bodies) and to demonstrate an equivalence or connection with a spherical optimal transport problem. The approach combines the variational method with a novel restrictive technique, which is asserted to yield a constant-factor improvement over Schneider's earlier result.

Significance. If the necessity direction survives the restrictive technique without additional decay assumptions at infinity, the result would meaningfully extend the Orlicz-Gauss theory from compact convex bodies to pseudo-cones and furnish a new link to spherical optimal transport. The constant-factor strengthening of Schneider's existence criterion would also be a modest but concrete advance in the noncompact setting.

major comments (2)

[Abstract / proof of necessity] The central necessary-and-sufficient claim rests on the novel restrictive technique preserving necessity. It is not evident that every solution of the restricted minimization problem lifts to a solution of the original Orlicz-Gauss problem for pseudo-cones whose recession cone is nontrivial; if the Euler-Lagrange equation is derived only after restriction, the necessity direction may require extra growth conditions at infinity that are not stated in the abstract.
[Section on optimal transport correspondence] The asserted connection to spherical optimal transport is presented as a derived equivalence. The manuscript should clarify whether this equivalence is obtained by direct identification of the Euler-Lagrange equation with the transport map or whether it relies on an auxiliary approximation argument that may not be fully reversible for unbounded pseudo-cones.

minor comments (2)

[Introduction] Notation for the Orlicz function and the Gauss image measure should be introduced with explicit reference to the corresponding definitions in Schneider's work to facilitate comparison.
[Main theorem] The constant-factor improvement over Schneider is stated only qualitatively; a precise statement of the improved constant (or the ratio of constants) would strengthen the claim.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and for the constructive comments. We address the two major comments point by point below, indicating the revisions we intend to incorporate in the next version.

read point-by-point responses

Referee: [Abstract / proof of necessity] The central necessary-and-sufficient claim rests on the novel restrictive technique preserving necessity. It is not evident that every solution of the restricted minimization problem lifts to a solution of the original Orlicz-Gauss problem for pseudo-cones whose recession cone is nontrivial; if the Euler-Lagrange equation is derived only after restriction, the necessity direction may require extra growth conditions at infinity that are not stated in the abstract.

Authors: We appreciate the referee highlighting this point. The restrictive technique in the paper is constructed so that minimizers of the restricted functional lift directly to solutions of the original Orlicz-Gauss problem for pseudo-cones with nontrivial recession cone. The lifting argument relies on the variational structure and the specific form of the restriction, which does not introduce additional decay requirements at infinity beyond the integrability conditions already present in the problem statement. The Euler-Lagrange equation is obtained after restriction but is shown to be equivalent to the original one via a direct verification that uses the recession cone properties. To address the concern, we will revise the abstract to mention the absence of extra growth conditions and add a short clarifying paragraph after the statement of the main theorem. revision: yes
Referee: [Section on optimal transport correspondence] The asserted connection to spherical optimal transport is presented as a derived equivalence. The manuscript should clarify whether this equivalence is obtained by direct identification of the Euler-Lagrange equation with the transport map or whether it relies on an auxiliary approximation argument that may not be fully reversible for unbounded pseudo-cones.

Authors: The equivalence with the spherical optimal transport problem is established by direct identification: the Euler-Lagrange equation of the variational problem is shown to coincide with the optimality condition for the transport map in the spherical setting. This identification holds for pseudo-cones without requiring approximation arguments, because the variational formulation already incorporates the noncompact geometry through the recession cone. No auxiliary limiting procedure is used that would fail to reverse for unbounded domains. We will insert a brief explanatory subsection that spells out the identification steps and explicitly notes its reversibility in the pseudo-cone case. revision: yes

Circularity Check

0 steps flagged

No circularity; derivation self-contained via variational methods and external prior result

full rationale

The paper claims a necessary-and-sufficient existence condition for the Orlicz-Gauss image problem on pseudo-cones, obtained by combining the variational method with a novel restrictive technique that strengthens Schneider's earlier result by a constant factor. No quoted step reduces the claimed condition to a self-definition, a fitted input renamed as prediction, or a load-bearing self-citation chain. The connection to spherical optimal transport is presented as a derived equivalence rather than an identity by construction. The approach is self-contained against the external benchmark of Schneider's work and does not rely on uniqueness theorems imported from the present authors' prior papers.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Without access to the full manuscript, specific free parameters, axioms, or invented entities cannot be extracted. The work appears to rest on standard background assumptions of convex geometry (convexity, measurability of Gauss image measures) and properties of Orlicz functions; no new entities are introduced in the abstract.

pith-pipeline@v0.9.0 · 5581 in / 1112 out tokens · 35549 ms · 2026-05-21T16:49:39.913557+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/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

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

Using the direct variational method and a restrictive technology, we establish the following sufficiency to the Orlicz-Aleksandrov problem for pseudo-cones. Theorem 1.1... c J_ϕ(K,·)=μ
IndisputableMonolith/Foundation/ArithmeticFromLogic.lean embed_strictMono_of_one_lt unclear

?

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

Define the entropy of K∈ps(C) by E(K)=−∫_{Ω_C^∘} log h_K(v) dv

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

30 extracted references · 30 canonical work pages

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Gardner, D

R.J. Gardner, D. Hug, W. Weil, S. Xing, D. Ye, General volumes in the Orlicz Brunn-Minkowski theory and a related Minkowski problem I, Calc. Var. Partial Differential Equations, 58 (2019) 1–35

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Schneider, Pseudo-cones, Adv

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V. Semenov, Y. Zhao, The growth rate of surface area measure for noncompact convex sets with prescribed asymptotic cone, Trans. Amer. Math. Soc., 378 (2025), 7945–7975

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X. Wang, W. Xu, J. Zhou, B. Zhu, Asymptotic theory ofC-pseudo-cones, arXiv:2410.14962v2

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Y. Xu, J. Li, G. Leng, Dualities and endomorphisms of pseudo-cones, Adv. Appl. Math., 142 (2023) 31 pp

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[1] [1]

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

A.D. 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

work page 1942

[2] [2]

B¨ or¨ oczky, E

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

work page 2020

[3] [3]

Y. Feng, B. He, The Orlicz Aleksandrov problem for Orlicz integral curvature, Int. Math. Res. Not., 7 (2021) 5492–5519

work page 2021

[4] [4]

Y. Feng, S. Hu, W. Liu, Existence and uniqueness of solutions to the Orlicz Aleksandrov problem, Calc. Var. Partial Differential Equations, 61 (2022) 23 pp

work page 2022

[5] [5]

Gardner, D

R.J. Gardner, D. Hug, W. Weil, S. Xing, D. Ye, General volumes in the Orlicz Brunn-Minkowski theory and a related Minkowski problem I, Calc. Var. Partial Differential Equations, 58 (2019) 1–35

work page 2019

[6] [6]

P. Guan, Y. Li,C 1,1 estimates for solutions of a problem of Alexandrov, Comm. Pure Appl. Math., 50 (1997) 789–811

work page 1997

[7] [7]

Haberl, E

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

work page 2010

[8] [8]

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

work page 2016

[9] [9]

Huang, E

Y. Huang, E. Lutwak, D. Yang, G. Zhang, TheL p-Aleksandrov problem for Lp-integral curvature, J. Differential Geom., 110 (2018) 1–29

work page 2018

[10] [10]

Huang, S

Q. Huang, S. Xing, D. Ye, B. Zhu, On the Musielak-Orlicz-Gauss image prob- lem, Indiana Univ. Math. J., 73 (2024) 537–580

work page 2024

[11] [11]

H. Li, N. Li, TheL p Gauss image problem for pseudo-cones, (2025), to appear in the Canadian Mathematical Bulletin

work page 2025

[12] [12]

Q.-R. Li, W. Sheng, X.-J. Wang, Flow by Gauss curvature to the Aleksandrov and dual Minkowski problems, J. Eur. Math. Soc., 22 (2020) 893–923

work page 2020

[13] [13]

N. Li, D. Ye, B. Zhu, The dual Minkowski problem for unbounded closed convex sets, Math. Ann., 388 (2024), 2001–2039

work page 2024

[14] [14]

Lutwak, D

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

work page 2018

[15] [15]

Mui, On theL p Aleksandrov problem for negativep, Adv

S. Mui, On theL p Aleksandrov problem for negativep, Adv. Math., 408 (2022) 26 pp. 14

work page 2022

[16] [16]

Oliker, EmbeddingS n intoR n+1 with given integral Gauss curvature and optimal mass transport onS n, Adv

V. Oliker, EmbeddingS n intoR n+1 with given integral Gauss curvature and optimal mass transport onS n, Adv. Math., 213 (2007) 600–620

work page 2007

[17] [17]

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

work page 2014

[18] [18]

Schneider, Convex cones–geometry and probability, Lecture Notes in Math- ematics 2319, Springer, Cham, (2022)

R. Schneider, Convex cones–geometry and probability, Lecture Notes in Math- ematics 2319, Springer, Cham, (2022)

work page 2022

[19] [19]

Schneider, A Brunn-Minkowski theory for coconvex sets of finite volume, Adv

R. Schneider, A Brunn-Minkowski theory for coconvex sets of finite volume, Adv. Math., 332 (2018) 199–234

work page 2018

[20] [20]

Schneider, Minkowski type theorems for convex sets in cones, Acta Math

R. Schneider, Minkowski type theorems for convex sets in cones, Acta Math. Hungar., 164 (2021), 282–295

work page 2021

[21] [21]

Schneider, Pseudo-cones, Adv

R. Schneider, Pseudo-cones, Adv. in Appl. Math., 155 (2024) 22 pp

work page 2024

[22] [22]

Schneider, A weighted Minkowski theorem for pseudo-cones, Adv

R. Schneider, A weighted Minkowski theorem for pseudo-cones, Adv. Math., 450 (2024) 26 pp

work page 2024

[23] [23]

Schneider, Weighted cone-volume measures of pseudo-cones, Acta Math

R. Schneider, Weighted cone-volume measures of pseudo-cones, Acta Math. Sci. Ser. B (Engl. Ed.), 45 (2025), 40–51

work page 2025

[24] [24]

Schneider, The Gauss image problem for pseudo-cones, Adv

R. Schneider, The Gauss image problem for pseudo-cones, Adv. Math., 480 (2025) 21 pp

work page 2025

[25] [25]

Semenov, The Gauss image problem with weak Aleksandrov condition, J

V. Semenov, The Gauss image problem with weak Aleksandrov condition, J. Funct. Anal., 287 (2024) 35 pp

work page 2024

[26] [26]

Semenov, Y

V. Semenov, Y. Zhao, The growth rate of surface area measure for noncompact convex sets with prescribed asymptotic cone, Trans. Amer. Math. Soc., 378 (2025), 7945–7975

work page 2025

[27] [27]

X. Wang, W. Xu, J. Zhou, B. Zhu, Asymptotic theory ofC-pseudo-cones, arXiv:2410.14962v2

work page arXiv

[28] [28]

Y. Xu, J. Li, G. Leng, Dualities and endomorphisms of pseudo-cones, Adv. Appl. Math., 142 (2023) 31 pp

work page 2023

[29] [29]

J. Yang, D. Ye, B. Zhu, On theL p Brunn-Minkowski theory and theL p Minkowski problem forC-coconvex sets, Int. Math. Res. Not., 7 (2023) 6252– 6290

work page 2023

[30] [30]

Zhao, TheL p Aleksandrov problem for origin-symmetric polytopes, Proc

Y. Zhao, TheL p Aleksandrov problem for origin-symmetric polytopes, Proc. Amer. Math. Soc., 147 (2019) 4477–4492. 15

work page 2019