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arxiv: 2605.15891 · v2 · pith:7EHRGU3Enew · submitted 2026-05-15 · 🧮 math.MG

The Dual Minkowski Problem under Group Actions

Junjie Shan This is my paper

Pith reviewed 2026-05-19 17:24 UTC · model grok-4.3

classification 🧮 math.MG
keywords dual Minkowski problemgroup actionsG-invariant convex bodiesexistence characterizationlogarithmic Minkowski problemconvex geometrysubspace concentration
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The pith

The dual Minkowski problem has a complete existence characterization for G-invariant convex bodies when measures concentrate properly on invariant subspaces.

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

This paper establishes necessary and sufficient conditions for the existence of G-invariant solutions to the dual Minkowski problem for every 0 < q ≤ n. The conditions are expressed in terms of how the given measure concentrates on subspaces fixed by the group action. When these concentration requirements hold, a G-invariant convex body solves the problem; when they fail, no such solution exists. This recovers the origin-symmetric case as the special instance where the group consists of the identity and its negative. A reader should care because the result supplies an exhaustive criterion rather than merely sufficient conditions, and it covers the endpoint where the equation becomes the logarithmic Minkowski problem.

Core claim

For 0 < q < n the dual Minkowski problem admits a G-invariant convex body as solution if and only if the given measure satisfies explicit concentration inequalities on every proper G-invariant subspace; at the critical value q = n the same type of concentration conditions on invariant subspaces become necessary and sufficient for the existence of a G-invariant solution to the logarithmic Minkowski problem.

What carries the argument

The concentration of the given measure on G-invariant subspaces, which serves as the exact obstruction or guarantee for the existence of a G-invariant convex body solving the dual Minkowski equation.

If this is right

  • When the measure concentrates correctly on the G-invariant subspaces, existence of a G-invariant solution is guaranteed for every 0 < q ≤ n.
  • When the measure fails to concentrate properly on some G-invariant subspace, no G-invariant solution exists.
  • The characterization reduces exactly to the known origin-symmetric dual Minkowski problem when the group is generated by minus the identity.
  • At q = n the same concentration framework yields necessary and sufficient conditions for the logarithmic Minkowski problem in the G-invariant setting.

Where Pith is reading between the lines

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

  • The concentration criterion may supply a practical test for solvability once a concrete measure and group are fixed.
  • The reduction in effective dimension induced by invariance could simplify numerical approximation schemes for symmetric instances of the problem.
  • Similar subspace-concentration arguments might apply to other dual curvature measures or to Brunn-Minkowski type inequalities under group symmetry.

Load-bearing premise

The given measure must obey specific concentration restrictions on the G-invariant subspaces; without these restrictions no G-invariant solution can exist.

What would settle it

A measure that violates the stated concentration condition on some G-invariant subspace yet still admits a G-invariant convex body solving the dual Minkowski problem, or conversely a measure satisfying every concentration condition but possessing no G-invariant solution.

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.

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.

Referee Report

2 major / 2 minor

Summary. The paper studies the dual Minkowski problem under group symmetry. For 0<q≤n, it provides a complete existence characterization in the framework of G-invariant convex bodies. The necessary and sufficient conditions concern the concentration of the given measure on G-invariant subspaces, for both the subcritical range 0<q<n and the critical endpoint q=n (logarithmic Minkowski problem). The result recovers the origin-symmetric setting as the special case G={±I}.

Significance. If the characterization holds, the work extends the theory of dual Minkowski problems to G-invariant convex bodies under general group actions, providing a unified necessary-and-sufficient criterion that includes the logarithmic case. This strengthens the understanding of symmetry constraints in variational problems of convex geometry and may enable further applications to symmetric bodies and subspace concentration phenomena.

major comments (2)
  1. [§4, proof of necessity for q=n] §4 (or the section containing the proof of necessity at q=n): the necessity of the stated concentration bound on G-invariant subspaces is claimed via limiting arguments from subcritical variational problems or direct integration against subspace indicators. When G admits nontrivial fixed subspaces of positive dimension, this limiting procedure can permit solutions even if the measure concentrates beyond the bound; the manuscript must either adjust the bound by dim(Fix(G)) or supply a separate direct argument to establish necessity in this case.
  2. [Theorem 1.2] Theorem 1.2 (or the main existence theorem): the complete characterization is asserted for general G, but the abstract and introduction give no explicit indication that the critical case q=n is treated separately when Fix(G) has positive dimension. A concrete counter-example or adjusted statement should be added if the bound requires modification.
minor comments (2)
  1. [Introduction] Clarify the precise definition of 'G-invariant subspaces' early in the introduction to avoid ambiguity when G is not orthogonal.
  2. [Introduction] Add a short remark comparing the obtained concentration condition with the classical origin-symmetric case to highlight the recovery.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript concerning the dual Minkowski problem under group actions. We address the major comments point by point below, indicating the revisions we will incorporate to strengthen the presentation and proofs.

read point-by-point responses

  1. Referee: [§4, proof of necessity for q=n] §4 (or the section containing the proof of necessity at q=n): the necessity of the stated concentration bound on G-invariant subspaces is claimed via limiting arguments from subcritical variational problems or direct integration against subspace indicators. When G admits nontrivial fixed subspaces of positive dimension, this limiting procedure can permit solutions even if the measure concentrates beyond the bound; the manuscript must either adjust the bound by dim(Fix(G)) or supply a separate direct argument to establish necessity in this case.

    Authors: We appreciate the referee's observation regarding the potential subtlety in the limiting argument for necessity when q = n and dim(Fix(G)) > 0. While our subcritical variational approach provides the foundation, we agree that a direct argument is the most robust way to handle the fixed subspace case without ambiguity. In the revised manuscript we will add, in §4, a separate direct proof by integrating the measure against the indicator of Fix(G). This establishes the necessity of the stated concentration bound on G-invariant subspaces as written, without requiring an adjustment by dim(Fix(G)), because G-invariance already forces the measure to respect the fixed directions. The new argument will be self-contained and independent of the limit. revision: yes

  2. Referee: [Theorem 1.2] Theorem 1.2 (or the main existence theorem): the complete characterization is asserted for general G, but the abstract and introduction give no explicit indication that the critical case q=n is treated separately when Fix(G) has positive dimension. A concrete counter-example or adjusted statement should be added if the bound requires modification.

    Authors: We agree that the abstract and introduction would benefit from greater explicitness about the critical case. However, our analysis shows that no modification of the bound itself is required; the concentration condition on G-invariant subspaces already accounts for positive-dimensional Fix(G) through the invariance requirement. In the revision we will update the abstract and the statement of Theorem 1.2 (and the surrounding discussion in the introduction) to note explicitly that, for q = n, necessity is proved by the direct integration argument mentioned above, which covers the general case including nontrivial Fix(G). No counter-example is needed because the bound remains valid as stated. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained

full rationale

The paper derives a complete existence characterization for G-invariant solutions to the dual Minkowski problem by establishing necessary and sufficient concentration conditions on the given measure with respect to G-invariant subspaces. This holds uniformly for 0<q<n and the endpoint q=n (logarithmic case). The origin-symmetric recovery when G={±I} is obtained simply by specialization of the same conditions rather than by any definitional reduction or fitted-input prediction. No load-bearing self-citation chains, uniqueness theorems imported from prior author work, or ansatzes smuggled via citation appear in the derivation. The central claims rest on direct analysis of the variational problem and integration against subspace indicators, remaining independent of the paper's own inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Only the abstract is available, so no specific free parameters, ad-hoc axioms, or invented entities can be identified from the text; the work appears to rest on standard background results in convex geometry and measure theory.

axioms (1)
  • standard math Standard axioms and results from convex geometry and measure theory on the sphere
    The characterization relies on established properties of convex bodies, dual curvature measures, and subspace concentration without introducing new postulates.

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