{"id":"be8358c3-d550-44bb-99c9-32a8b501882a","arxiv_id":"2607.04418","paper_version":1,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":7.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"For every j≥2 and N≥2, if f∘λ_j is concave on convex bodies then f must be constant.","lead":"Higher Dirichlet eigenvalues of convex bodies admit no nontrivial Brunn-Minkowski concavity, even after any reparametrization by a scalar function f. The result shows that the classical Brascamp-Lieb inequality for the first eigenvalue is special and cannot extend.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.5","headline":"No significant objection identified","rationale":"The reader correctly isolates the mode-ordering constructions as the only non-classical step and correctly judges them to be verified by direct computation. Independent inspection of the same lemmas confirms that the algebraic identities (6), the sign of the first-order Taylor coefficient of the midpoint side length, and the continuity argument of Lemma 4 together produce a genuine open interval of ratios about 1. The product construction of Proposition 7 is likewise elementary once the planar case is settled. Consequently the impossibility statement of Theorem 3 stands, and the ACCEPT verdict with high confidence requires no adjustment.","tokens_in":10458,"tokens_out":454,"duration_ms":4987,"concrete_test":"For a concrete j (e.g. j=2) recompute the first several normalized eigenvalues of R_(A,B), of the two perturbed rectangles K_δ and L_δ for a small positive δ, and of their midpoint M_δ; verify that the j-th value remains exactly 1 on the endpoints and strictly exceeds 1 on the midpoint, confirming that the intermediate-value argument of Lemma 5 produces every ratio in [1,r_+].","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim of Theorem 3 rests on the rectangular constructions of Lemmas 5–6 and their product extension in Proposition 7. Those constructions are elementary and fully explicit: the base rectangle R_(A,B) is chosen so that the (j,1) and (1,2) modes coincide at the j-th eigenvalue while all lower modes lie strictly below; the two one-parameter families φ_v and φ_h preserve that value; Taylor expansion of the midpoint side length shows it drops below B, forcing the midpoint eigenvalue strictly above π^{2}; continuity of ν_j (Lemma 4) then fills a whole interval of ratios around 1. The same mode-ordering argument, together with a large enough cube factor, lifts the construction to every dimension N≥2. Every inequality used is a direct comparison of explicit algebraic expressions, so the weakest link identified by the reader is in fact secure. No hidden assumption or gap appears.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.5","summary":"The paper proves that for every j ≥ 2 and N ≥ 2, the only functions f : (0, ∞) \to ℝ for which K ↦ (f ∘ λ_j)(K) is concave on the class of convex bodies in ℝ^N are the constant functions (Theorem 3). The argument proceeds by constructing, for every ratio σ in a neighborhood of 1, pairs of rectangular boxes with equal j-th eigenvalues whose Minkowski average has j-th eigenvalue exactly σ times larger (or smaller); scaling and iteration then force f(σx) = f(x) for all x > 0, hence constancy. As a byproduct, Proposition 9 characterises the functions f for which f ∘ λ_1 is concave: they are precisely those of the form f(r) = g(r^{-1/2}) with g concave and non-decreasing.","tokens_in":10639,"tokens_out":782,"duration_ms":42052,"significance":"The result cleanly closes a natural question left open by the negative example of Bucur–Fragalà–Lamboley for the specific map λ_2^{-1/2} in dimension 2: no scalar reparametrisation can restore Brunn–Minkowski concavity for any higher Dirichlet eigenvalue. The proof is elementary, fully explicit, and relies only on the classical spectrum of rectangles together with a product construction that lifts the planar case to every dimension N ≥ 2. The same technique yields a sharp characterisation of the admissible reparametrisations for the first eigenvalue, extending the classical Brascamp–Lieb inequality. The paper is short, self-contained, and free of hidden analytic machinery, which makes the obstruction transparent and robust.","major_comments":[],"minor_comments":[{"comment":"Title: “higher Dirichlet eigenvalue” should be plural (“eigenvalues”) for consistency with the abstract and the body of the paper.","section":null},{"comment":"Lemma 4: the continuity argument is written only for planar rectangles; a one-line remark that the same isolation-of-modes argument works for rectangular boxes in any dimension would make the later product construction slightly cleaner (though it is not logically required).","section":null},{"comment":"Proposition 7: the displayed formula for r_0 is broken across lines in a way that makes the second term ambiguous; rewriting it as r_0 = min{1+(r_+-1)/(1+r),(1+r)/(r+r_-)} would remove any possible misreading.","section":null},{"comment":"Page 9, proof of Theorem 3: the sentence “Sets:=pλ_j(K)/x” is missing spaces and a square-root symbol in the extracted text; a quick proof-reading pass will catch similar extraction artefacts.","section":null},{"comment":"A short remark after Proposition 7 noting that the same pairs already live inside the subclass of rectangular parallelepipeds would emphasise that the obstruction is even stronger than stated.","section":null}],"recommendation":"accept","confidential_remarks":"The manuscript is short, elementary and decisive; it is a good fit for a geometry or analysis journal that values clean negative results. No concerns about novelty, citations or scope."},"author_rebuttal":null,"desk_editor":{"model":"grok-4.5","letter":"This paper settles a natural question left open by Bucur–Fragalà–Lamboley: the failure of Brunn–Minkowski for higher Dirichlet eigenvalues is not an artifact of the map s↦s^{-1/2}. For every j≥2 and N≥2, if f∘λ_j is concave on convex bodies then f is constant (Theorem 3). They also give a clean characterization for λ_1: f∘λ_1 is concave iff g(r)=f(r^{-2}) is concave and non-decreasing (Proposition 9).\n\nWhat is new is the arbitrary-reparametrization statement and the uniform treatment of all j and N. The earlier 2-D counter-example for the specific homogeneous map is recovered and strengthened. The argument is elementary and fully written. They pick a base rectangle where the (j,1) and (1,2) modes coincide at the j-th eigenvalue while lower modes sit strictly below, then deform along two explicit curves φ_v and φ_h that keep λ_j fixed. Taylor expansion shows the midpoint side length drops, so the midpoint eigenvalue jumps above the common value; continuity of \nu_j on rectangles fills an interval of ratios around 1. A thin cube factor lifts the construction to higher dimensions without disturbing the first j eigenvalues. All comparisons are direct algebraic inequalities; there are no free parameters or hidden analytic estimates.\n\nThe soft spots are minor. The mode-ordering claims are verified by hand for the chosen A,B, and the continuity lemma is standard. The result is confined to the convex-body setting and does not address non-convex domains or other operators, but that is outside the stated scope. Citations are appropriate and the classical background facts are used correctly.\n\nThis is for people working on Brunn–Minkowski inequalities for variational functionals or spectral shape optimization. The math is solid and self-contained; a serious referee will verify the mode counts in a few pages and accept. I would send it to peer review without hesitation.","headline":"Clean impossibility result: no non-constant f makes f∘λ_j concave for j≥2, via explicit rectangular constructions that work in every dimension.","tokens_in":11252,"tokens_out":556,"would_cite":true,"duration_ms":8015,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["52A20","47A10","35J15","39B62"],"pacs":[],"model":"grok-4.5","headline":"Higher Dirichlet eigenvalues admit no nontrivial Brunn–Minkowski concavity: any reparametrization that works must be constant.","keywords":["convex body","Brunn–Minkowski inequality","Dirichlet eigenvalues","concavity","higher eigenvalues","rectangular boxes"],"falsifier":"Exhibit a non-constant continuous f and a pair of convex bodies in dimension ≥ 2 for which f(λ_j) fails the midpoint inequality for some j ≥ 2, or prove that no such rectangles exist that realize every ratio σ near 1.","tokens_in":11363,"feed_emoji":"📐","tokens_out":919,"duration_ms":8093,"temperature":0.7,"pith_summary":"The first Dirichlet eigenvalue of a convex body satisfies a classical Brunn–Minkowski inequality: the map that sends a body to the reciprocal square root of that eigenvalue is concave under Minkowski combinations. The same scaling suggests that higher eigenvalues might obey an analogous inequality after a suitable reparametrization. This paper proves they cannot. For every index j at least 2 and every dimension at least 2, if a scalar function of the j-th eigenvalue is concave on the family of all convex bodies, then that function is forced to be constant. The argument constructs pairs of rectangular boxes that share the same j-th eigenvalue while the j-th eigenvalue of their average can be made either larger or smaller, then scales the construction so that every ratio near 1 is realized. The same technique yields a clean characterization of precisely which reparametrizations of the first eigenvalue remain concave.","feed_headline":"Higher eigenvalues kill Brunn–Minkowski concavity","feed_subtitle":"Any reparametrization of λ_j for j≥2 that stays concave on convex bodies must be constant.","key_machinery":"A rectangular construction (Lemmas 5–6 and Proposition 7) that produces pairs of boxes K, L with λ_j(K) = λ_j(L) while λ_j((K + L)/2) = σ λ_j(K) for every ratio σ in a neighborhood of 1; continuity of the j-th eigenvalue on side lengths then forces any concave reparametrization to be constant.","core_discovery":"For any j ≥ 2 and N ≥ 2, if K ↦ (f ∘ λ_j)(K) is concave on the family of convex bodies in R^N for some function f : (0, ∞) → R, then f must be constant. In other words, no nontrivial scalar reparametrization restores Brunn–Minkowski concavity for higher Dirichlet eigenvalues.","pith_inferences":["The same rectangular technique may obstruct concavity statements for other spectral quantities that share the same scaling, such as higher Robin or Neumann eigenvalues on convex bodies.","Once rectangles alone force constancy, any larger class containing rectangles inherits the same impossibility, so the result is robust under domain enlargement.","The characterization for λ_1 suggests that monotonicity of the reparametrization, not merely concavity, is the feature that distinguishes the first eigenvalue from the rest."],"forward_implications":["Any search for Brunn–Minkowski-type inequalities for λ_j with j ≥ 2 on convex bodies is necessarily empty once an arbitrary reparametrization is allowed.","The only concave reparametrizations of λ_1 are those of the form f(r) = g(r^{-1/2}) where g is concave and non-decreasing.","The obstruction already appears for planar rectangles and therefore persists after product extension to higher dimensions.","Results that treat higher eigenvalues must either restrict the class of domains or abandon pure concavity under Minkowski combination."],"fun_headline_variants":["No nontrivial reparametrization restores BM concavity for λ_j j≥2","Higher Dirichlet eigenvalues admit only constant concave transforms","Any f∘λ_j concave on convex bodies forces f constant when j≥2","Brunn-Minkowski fails for all higher eigenvalues under any reparametrization","Nontrivial BM inequality impossible for λ_j with j≥2 in any dimension"],"cache_read_input_tokens":128,"weakest_assumption_plain":"The explicit rectangles used in the construction really do keep the first j modes ordered so that the j-th eigenvalue can be prescribed independently of the two endpoints while their midpoint ratio sweeps a full interval around 1.","fun_headline_variants_meta":{"raw":{"variants":["No nontrivial reparametrization restores BM concavity for λ_j j≥2","Higher Dirichlet eigenvalues admit only constant concave transforms","Any f∘λ_j concave on convex bodies forces f constant when j≥2","Brunn-Minkowski fails for all higher eigenvalues under any reparametrization","Nontrivial BM inequality impossible for λ_j with j≥2 in any dimension"]},"model":"grok-4.5","effort":"low","cost_usd":0.006202,"raw_usage":{"total_tokens":1559,"prompt_tokens":692,"num_sources_used":0,"completion_tokens":103,"cost_in_usd_ticks":62020000,"prompt_tokens_details":{"text_tokens":692,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":764,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":692,"tokens_out":103,"duration_ms":6614,"temperature":1.0,"reasoning_tokens":764,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-11T19:18:21.269055+00:00","model_set":{"reader":"grok-4.5"},"falsifier":"Exhibit a non-constant continuous f and a pair of convex bodies in dimension ≥ 2 for which f(λ_j) fails the midpoint inequality for some j ≥ 2, or prove that no such rectangles exist that realize every ratio σ near 1.","supporting_citations":[],"review_version":1}