{"id":"25a9496a-4935-437d-93f7-9932f372f7ed","arxiv_id":"2605.15891","paper_version":2,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":6.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"The paper gives necessary and sufficient conditions on a measure for the existence of G-invariant convex bodies solving the dual Minkowski problem for 0 < q ≤ n when G ⊂ O(n) has no nonzero fixed points.","lead":"The paper provides a complete existence characterization for solutions to the dual Minkowski problem that are invariant under a group G of orthogonal transformations with no nonzero fixed points. A smart generalist might read it to see how symmetry constraints translate into precise conditions on measures for existence in convex geometry, including the logarithmic case.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.3","headline":"No significant objection identified","rationale":"Reader correctly flags UNVERDICTED status due to missing full text. With only the abstract available, no load-bearing technical flaw in the argument can be identified or tested; the structural hypothesis on G is presented as the enabling assumption rather than a hidden one.","tokens_in":1609,"tokens_out":242,"duration_ms":16568,"concrete_test":"Locate and read the statement of the main existence theorem (likely Theorem 1.1 or equivalent) together with the definition of the concentration conditions on G-invariant subspaces; confirm that the necessity and sufficiency statements match the abstract claim for both 0<q<n and q=n.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The provided context contains only the abstract and a placeholder reference to full text; no derivations, theorems, or proofs are available for inspection. The central claim is an existence characterization for G-invariant bodies under the stated hypothesis on G, recovering the origin-symmetric case. The no-nonzero-fixed-points condition is explicitly invoked to obtain the necessary concentration criteria on invariant subspaces, and no internal inconsistency is detectable from the abstract alone.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","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.","tokens_in":1659,"tokens_out":296,"duration_ms":23539,"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":[{"comment":"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.","section":"Abstract"}],"minor_comments":[],"recommendation":"uncertain","confidential_remarks":"Only the abstract was accessible; the full manuscript text referenced in the query was not supplied, which directly accounts for the low soundness rating and uncertain recommendation."},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for reviewing our manuscript on the dual Minkowski problem under group actions. We address the single major comment below.","responses":[{"response":"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_made":"no","referee_comment":"[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."}],"tokens_in":1140,"tokens_out":207,"duration_ms":25359,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","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.\n\nWhat 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.\n\nThe 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.\n\nThe 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.\n\nThis 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.","headline":"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.","tokens_in":2138,"tokens_out":364,"would_cite":false,"duration_ms":18026,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.3","headline":"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.","keywords":["dual Minkowski problem","G-invariant convex bodies","group actions","logarithmic Minkowski problem","existence characterization","orthogonal group","convex geometry","subspace concentration"],"falsifier":"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.","tokens_in":2482,"feed_emoji":"📐","tokens_out":635,"duration_ms":21136,"temperature":0.7,"pith_summary":"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.","feed_headline":"Group symmetry gives complete existence conditions for dual Minkowski problem","feed_subtitle":"Measure concentration on invariant subspaces determines solvability for all q up to n when G has no fixed points","key_machinery":"Concentration conditions of the measure on G-invariant subspaces, which supply the necessary and sufficient criteria for existence.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"fun_headline_variants":["G-symmetry determines dual Minkowski existence for all q ≤ n","Measure concentration on G-subspaces solves dual Minkowski problem","Dual Minkowski admits solutions iff measures concentrate on subspaces","Group actions characterize dual Minkowski up to the q=n case"],"cache_read_input_tokens":2112,"weakest_assumption_plain":"The group G must have no nonzero fixed vectors so that the relevant concentration statements on invariant subspaces can be derived.","fun_headline_variants_meta":{"raw":{"variants":["G-symmetry determines dual Minkowski existence for all q ≤ n","Measure concentration on G-subspaces solves dual Minkowski problem","Dual Minkowski admits solutions iff measures concentrate on subspaces","Group actions characterize dual Minkowski up to the q=n case"]},"model":"grok-4.3","cost_usd":0.003466,"raw_usage":{"total_tokens":1752,"prompt_tokens":516,"num_sources_used":0,"completion_tokens":63,"cost_in_usd_ticks":34662000,"prompt_tokens_details":{"text_tokens":516,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":1173,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":516,"tokens_out":63,"duration_ms":11745,"temperature":1.0,"reasoning_tokens":1173,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-06-30T19:36:27.571366+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"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.","supporting_citations":[],"review_version":2}