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The Dual Minkowski Problem under Group Actions

T0 review · 1 major / 0 minor · reviewed 2026-06-30 · grok-4.3

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Pith's one-line read 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.

desk verdict 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 →

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

Junjie Shan This is my paper · ORCID
classification math.MG
keywords dualMinkowskiproblemG-invariantconvexbodiesgroupactionslogarithmicexistencecharacterizationorthogonalgeometrysubspaceconcentration
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 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.

What carries the argument

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

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.

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Extended reading notes

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.

Load-bearing premise

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

Editorial extensions

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.

Reading between the lines

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.
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Editorial analysis

A structured set of objections, weighed in public.

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

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)
  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

  1. 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 · score 0.0 of 10

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.

Assumptions & free parameters 0 free parameters · 1 assumptions · 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.

assumptions (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.

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Cite this review

Pith. "Pith review of The Dual Minkowski Problem under Group Actions." pith.science (2026). https://pith.science/paper/7EHRGU3E

@misc{pith2026260515891,
  author       = {Pith},
  title        = {Pith review of: The Dual Minkowski Problem under Group Actions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7EHRGU3E}},
  note         = {Machine review of arXiv:2605.15891}
}
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.

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Works this paper leans on

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