The Dual Minkowski Problem under Group Actions

Junjie Shan

For groups G in O(n) without nonzero fixed points, the dual Minkowski problem has G-invariant solutions exactly when the data measure concentrates on G-invariant subspaces.

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

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2026-06-30 19:36 UTC pith:7EHRGU3E

load-bearing objection The paper claims a complete existence characterization for the dual Minkowski problem on G-invariant bodies when G has no fixed points, recovering the origin-symmetric case, but the abstract alone leaves the proof uncheckable. the 1 major comments →

arxiv 2605.15891 v2 pith:7EHRGU3E submitted 2026-05-15 math.MG

The Dual Minkowski Problem under Group Actions

Junjie Shan This is my paper

classification math.MG

keywords dual Minkowski problemG-invariant convex bodiesgroup actionslogarithmic Minkowski problemexistence characterizationorthogonal groupconvex geometrysubspace concentration

verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The paper establishes necessary and sufficient conditions for existence of solutions to the dual Minkowski problem restricted to convex bodies invariant under a group G acting orthogonally. These conditions are expressed in terms of how a given measure concentrates on the G-invariant linear subspaces, and they hold for every exponent q between 0 and n inclusive. When the group is simply {I, -I}, the result specializes to the already-known origin-symmetric case. The endpoint q = n recovers a symmetric version of the logarithmic Minkowski problem. A reader cares because the characterization is complete: it identifies precisely which data admit symmetric solutions and which do not.

Core claim

For 0 < q ≤ n, in the class of G-invariant convex bodies, the dual Minkowski problem admits a solution if and only if the given measure satisfies explicit concentration conditions on every G-invariant subspace; at the critical value q = n the same conditions characterize solvability of the logarithmic Minkowski problem under the same symmetry.

What carries the argument

Concentration conditions of the measure on G-invariant subspaces, which supply the necessary and sufficient criteria for existence.

Load-bearing premise

The group G must have no nonzero fixed vectors so that the relevant concentration statements on invariant subspaces can be derived.

What would settle it

Exhibit a measure that violates one of the stated concentration conditions on a G-invariant subspace yet still admits a G-invariant solution body, or conversely a measure that obeys every listed condition but possesses no G-invariant solution.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

When G equals {±I} the new conditions reduce exactly to the known origin-symmetric dual Minkowski theorem.
At q = n the same subspace-concentration criterion solves the logarithmic Minkowski problem in the G-invariant setting.
The characterization is uniform across the open interval 0 < q < n and the endpoint q = n.
Uniqueness questions for the G-invariant solutions remain outside the existence statement.

Where Pith is reading between the lines

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

The same concentration language may apply to other Minkowski-type problems once an appropriate group action is fixed.
Computational searches for symmetric bodies could be restricted a priori to data satisfying the subspace conditions.
The result suggests that fixed-point-free orthogonal actions form a natural setting in which symmetry reduces the classical problem without losing solvability criteria.

Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

The paper claims a complete existence characterization for the dual Minkowski problem on G-invariant bodies when G has no fixed points, recovering the origin-symmetric case, but the abstract alone leaves the proof uncheckable.

read the letter

The core result is a necessary-and-sufficient condition on measure concentration on G-invariant subspaces that guarantees existence of G-invariant solutions to the dual Minkowski problem for 0 < q ≤ n. It treats the origin-symmetric case as the special instance G = {±I} and handles both the subcritical range and the logarithmic endpoint at q = n.

What is new is the move from the fixed ±I symmetry to a broader class of orthogonal groups satisfying the no-nonzero-fixed-points hypothesis. The concentration conditions look like a direct and reasonable generalization of the known symmetric criteria.

The paper does a clean job of stating the framework and the recovery of the classical case. If the arguments go through, the result supplies a usable benchmark for anyone working on symmetric versions of these problems.

The obvious limitation is that only the abstract is in front of us, so there is no way to inspect the derivations, check for gaps in the concentration arguments, or see how the no-fixed-points assumption is used in the estimates. That makes any judgment on soundness provisional.

This is for specialists in convex geometry who already follow Minkowski-type problems. A reader outside that narrow circle will not get much from it. I would send it to referees; the claim is concrete enough that a specialist can check it in reasonable time, and a positive verification would add a modest but solid extension to the literature.

Referee Report

1 major / 0 minor

Summary. The paper studies the dual Minkowski problem under group symmetry. For 0<q≤n, it claims a complete existence characterization for G-invariant convex bodies when G⊂O(n) has no nonzero fixed points. The necessary and sufficient conditions are stated in terms of concentration of the measure on G-invariant subspaces. This recovers the origin-symmetric setting when G={±I}. At q=n the problem reduces to the logarithmic Minkowski problem.

Significance. If the claimed characterization is correct, the work would extend the dual Minkowski problem to a natural class of symmetric settings, supplying necessary-and-sufficient conditions that generalize the origin-symmetric case. The explicit recovery of the G={±I} result provides a useful consistency check. The absence of free parameters in the stated conditions is a structural strength.

major comments (1)

[Abstract] Abstract: the central claim is a complete necessary-and-sufficient characterization, yet the provided text contains only the abstract statement with no derivations, theorems, or proof steps. This creates a verification gap that prevents assessment of whether the concentration conditions on G-invariant subspaces are indeed necessary and sufficient.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for reviewing our manuscript on the dual Minkowski problem under group actions. We address the single major comment below.

read point-by-point responses

Referee: [Abstract] Abstract: the central claim is a complete necessary-and-sufficient characterization, yet the provided text contains only the abstract statement with no derivations, theorems, or proof steps. This creates a verification gap that prevents assessment of whether the concentration conditions on G-invariant subspaces are indeed necessary and sufficient.

Authors: The abstract serves only as a concise overview of the main result. The full manuscript contains the precise theorem statements (necessary and sufficient conditions on the measure in terms of its concentration on proper G-invariant subspaces, for both 0<q<n and the endpoint q=n) together with complete proofs. These appear after the abstract in the submitted text, so the derivations and verification steps are available for assessment. revision: no

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper states a complete existence characterization for the dual Minkowski problem in the G-invariant setting under the explicit hypothesis that G has no nonzero fixed points, with necessary and sufficient conditions given directly as concentration properties of the measure on G-invariant subspaces. No equations, fitted parameters, or self-citations are exhibited that would reduce the claimed result to a redefinition of its inputs or to a prediction forced by construction. The recovery of the origin-symmetric case when G={±I} is presented as a special instance of the general framework rather than a tautology. The derivation chain therefore remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The result rests on the standard background theory of convex bodies and dual curvature measures together with one explicit domain assumption on the group.

axioms (1)

domain assumption G ⊂ O(n) is a subgroup with no nonzero fixed points
This hypothesis is required for the concentration conditions on G-invariant subspaces to serve as the necessary and sufficient criteria.

pith-pipeline@v0.9.1-grok · 5602 in / 1239 out tokens · 36774 ms · 2026-06-30T19:36:27.571366+00:00 · methodology

0 comments

read the original abstract

In this paper, we study the dual Minkowski problem under group symmetry. For $0<q\le n$, we give a complete existence characterization in the framework of $G$-invariant convex bodies when the group $G\subset O(n)$ has no nonzero fixed points, recovering the origin-symmetric setting when $G=\{\pm I\}$. The necessary and sufficient conditions concern the concentration of the measure on $G$-invariant subspaces, both in the range $0<q<n$ and at the critical endpoint $q=n$, where the problem becomes the logarithmic Minkowski problem.

Review history (2 revisions) →

discussion (0)

Reference graph

Works this paper leans on

46 extracted references · 46 canonical work pages

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B¨ or¨ oczky, E

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

work page 2013

[2] [2]

B¨ or¨ oczky, F

K.J. B¨ or¨ oczky, F. Fodor,The Lp dual Minkowski problem for p > 1 and q > 0, J. Differential Equations 266 (2019), 7980–8033

work page 2019

[3] [3]

B¨ or¨ oczky, P

K.J. B¨ or¨ oczky, P. Heged˝ us, G. Zhu,On the discrete logarithmic Minkowski prob- lem, Int. Math. Res. Not. 6 (2016), 1807–1838

work page 2016

[4] [4]

B¨ or¨ oczky, M

K.J. B¨ or¨ oczky, M. Henk,Cone-volume measure of general centered convex bodies, Adv. Math. 286 (2016), 703–721

work page 2016

[5] [5]

B¨ or¨ oczky, M

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

work page 2018

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B¨ or¨ oczky, E

K.J. B¨ or¨ oczky, E. Lutwak, D. Yang, G. Zhang,Affine images of isotropic mea- sures, J. Differential Geom. 99 (2015), 407–442

work page 2015

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B¨ or¨ oczky, E

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

work page 2019

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B¨ or¨ oczky,´A

K.J. B¨ or¨ oczky,´A. Kov´ acs, S. Mui, G. Zhang, Dual curvature density equation with group symmetry , J. Differential Equations 465 (2026), Paper No. 114197

work page 2026

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Bianchi, K.J

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

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

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

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Chen, Q.-R

H. Chen, Q.-R. Li, The Lp dual Minkowski problem and related parabolic flows , J. Funct. Anal. 281 (2021), Paper No. 109139

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S. Chen, Q.-R. Li, On the planar dual Minkowski problem , Adv. Math. 333 (2018), 87–117

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B. He, G. Leng, K. Li, Projection problems for symmetric polytopes, Adv. Math. 207 (2006), 73–90. 44 JUNJIE SHAN

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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

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Huang, E

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

work page 2018

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Huang, J

Y. Huang, J. Liu, D. Xi, Y. Zhao, Dual curvature measures for log-concave func- tions, J. Differential Geom. 128 (2024), 815–860

work page 2024

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Huang, Y

Y. Huang, Y. Zhao, On the Lp dual Minkowski problem, Adv. Math. 332 (2018), 57–84

work page 2018

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H. Jian, J. Lu, G. Zhu, Mirror symmetric solutions to the centro-affine Minkowski problem, Calc. Var. Partial Differential Equations 55 (2016), Art. 41, 22 pp

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Lutwak, D

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

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Lutwak, D

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

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Lutwak, D

E. Lutwak, D. Xi, D. Yang, G. Zhang, Chord measures in integral geometry and their Minkowski problems , Commun. Pure Appl. Math. 77 (2024), 3277–3330

work page 2024

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

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

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Q.-R. Li, J. Liu, J. Lu, Non-uniqueness of solutions to the dual Lp-Minkowski problem, Int. Math. Res. Not. (2022), 9114–9150

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J. Shan, The Lp dual Minkowski problem for group-invariant convex bodies, Adv. Math. 498 (2026), Paper No. 111038. THE DUAL MINKOWSKI PROBLEM UNDER GROUP ACTIONS 45

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Y. Zhao, Existence of solutions to the even dual Minkowski problem , J. Differen- tial Geom. 110 (2018), 543–572

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