{"id":"1050c611-a558-4416-b0ed-e41afdb8fb7f","arxiv_id":"2511.02688","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"No volume-constrained area maximizer among λ-convex bodies has a C² boundary, and every C² piece of such a maximizer has smallest principal curvature exactly λ, in constant-curvature space forms.","lead":"This paper shows that volume-constrained area maximizers among convex shapes whose curvature cannot go below a fixed value ('λ-convex' bodies) can never have a fully smooth boundary; wherever the boundary is smooth, the smallest curvature must sit exactly at the allowed minimum. The result is proved in Euclidean, spherical, and hyperbolic space using rolling-ball geometry plus classical stability theory for constant-mean-curvature hypersurfaces.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Hyperbolic λ≤1 case: Prop 3.4 excludes horosphere CMC pieces, yet the proof of Theorem 1.2(i) needs stability for such pieces; Prop 3.5 is stated only for λ∈I_Σ, leaving the full hyperbolic claim unproven.","rationale":"I read the paper in good faith. The Euclidean and spherical parts are well-structured, and the variational strategy is plausible. The most load-bearing weak point is indeed the hyperbolic λ≤1 case: Proposition 3.4 explicitly excludes horospheres, Proposition 3.5 is formally stated only for λ∈I_Σ, and Theorem 1.2(i) claims all λ>0. A horosphere patch has principal curvature 1, so for λ<1 it is a legitimate strictly λ-convex C² CMC piece; the written proof gives no stability argument for it. The likely fix is simple (on a horosphere the stability operator is the Euclidean Laplacian, giving positive second variation for zero-mean variations), but until that computation is included or the theorem statement is restricted, the full hyperbolic claim is not established by the written arguments. The reader's weakest assumption identifies the same issue, so I agree. I do not see a reason to reject the paper; the appropriate action is a conditional acceptance pending this technical clarification.","tokens_in":7856,"tokens_out":21827,"duration_ms":233835,"concrete_test":"Extend Prop 3.4 to horospheres in H^{n+1}: for a horosphere piece, compute the stability operator T=Δ and verify that for every nonconstant zero-mean f∈C_c^∞, -∫ f T(f) = ∫ |∇f|² > 0. Then run Prop 3.5, Case 2 with Ω an open horosphere patch for 0<λ<1, and check whether the volume-preserving second variation of area is positive. If it is, the hyperbolic λ≤1 gap is closed; if not, Theorem 1.2(i) must be restricted to λ∈I_H or the proof needs a different argument.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The main result claims Theorem 1.2(i) for all λ>0 in H^{n+1}, but the local improvement Proposition 3.5 is stated only for λ∈I_H=(1,∞), and its Case 2 relies on strong stability of the C² constant-mean-curvature piece. That stability is supplied by Proposition 3.4 only under the hypothesis that the piece is not contained in a horosphere. For λ<1 this hypothesis is not guaranteed: a C² λ-convex body can contain a patch of a horosphere (all principal curvatures 1>λ), which is a CMC, strictly λ-convex open set. Corollary 3.3 does produce a strictly λ-convex C² patch in this regime, but it does not rule out the patch being a horosphere. No stability computation for horosphere patches appears anywhere in the paper. If such a patch occurs, the second variation step in Proposition 3.5, as written, has no justification, so the proof of Theorem 1.2(i) does not cover λ≤1 in hyperbolic space. The asserted preservation of λ-convexity under the constructed variation in Prop 3.5 is also given without a detailed proof, but the horosphere stability gap is the decisive missing link.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the reverse isoperimetric problem in simply connected space forms: for a compact λ-convex body K of fixed volume maximizing perimeter, it claims (i) ∂K cannot be C², and (ii) on C² portions of a maximizer, the smallest principal curvature is identically λ (for λ in the range where geodesic spheres of curvature λ exist). The proof uses a rolling-ball theorem (Theorem 2.6) to show that every nontrivial λ-convex body lies in a nontrivial lens (Lemma 3.1), which is contained in a strictly smaller ball (Lemma 3.2); comparison then yields a strictly λ-convex C² patch (Corollary 3.3). The authors then construct a volume-preserving variation supported in such a patch (Proposition 3.5) which increases area, using the first variation when mean curvature is nonconstant and the second variation plus strong stability when it is constant. They conclude no C² maximizer can exist and the smallest curvature on C² pieces must be λ.","tokens_in":8213,"tokens_out":16988,"duration_ms":192784,"significance":"If correct, the result advances the reverse isoperimetric problem in dimension ≥3, where explicit maximizers are not generally known. The two properties — non-existence of smooth maximizers and saturation of the smallest principal curvature — are natural and elegant. The proof strategy is transparent and builds on established rolling and stability results; no parameters are fitted. The Euclidean and spherical parts are convincing. The hyperbolic λ≤1 case, however, is not fully justified as written; because the gap is localized and likely repairable, the core contribution remains valuable.","major_comments":[{"comment":"Theorem 1.2(i) states a result for all λ>0 in H^{n+1}, but the proof invokes Proposition 3.5, which is explicitly stated only for λ∈I_Σ (in the hyperbolic case λ>1). For 0<λ≤1, Corollary 3.3 provides a strictly λ-convex C² patch (the touching-sphere argument), but Proposition 3.5 cannot be applied in its stated form. No alternative variation argument is supplied. This leaves the main claim for hyperbolic λ≤1 unproved.","section":"Theorem 1.2 / Proposition 3.5"},{"comment":"Even if Proposition 3.5 is extended to λ≤1, Case 2 needs the strict inequality -∫_Ω vT(v)>0 for compactly supported zero-mean v on the constant-mean-curvature patch. The only stability criterion provided, Proposition 3.4, explicitly excludes hyperbolic CMC hypersurfaces contained in a horosphere. A C² λ-convex body with λ<1 may have a boundary portion that is a genuine horosphere patch (all principal curvatures 1>λ); Corollary 3.3 does not rule this out. Since no stability computation for horosphere patches appears, the proof of the hyperbolic case is incomplete. (A repair is straightforward: on a horosphere T=Δ, so -∫vT(v)=∫|∇v|²>0, or one can use the dilation Killing field y∂_y in the upper half-space model to extend Proposition 3.4.)","section":"Proposition 3.4 / Proposition 3.5, Case 2"},{"comment":"The admissibility of the constructed variation is asserted in one sentence ('as can be checked locally from Definition 1.1 using a case distinction'). This is a load-bearing point: the whole argument rests on K_{t,s} remaining λ-convex. Please give the detailed proof that a sufficiently small C² deformation supported in a strictly λ-convex patch preserves the λ-convexity condition, including at the boundary of the support.","section":"Proposition 3.5, first paragraph"}],"minor_comments":[{"comment":"The title block shows 'TWO PROPER TIES OF OPTIMISERS'; this should read 'TWO PROPERTIES OF OPTIMISERS'.","section":"Title page"},{"comment":"The definition τ:=|γ˙|²t is dimensionally inconsistent and the subsequent solution β(t)=a cos τ + b sin τ is hard to follow. It appears τ should be |γ˙|t. Please correct and clarify the reparameterization.","section":"Lemma 3.2, spherical case"},{"comment":"For hyperbolic λ≤1, RΣ(λ) is not defined, but Theorem 1.2(i) covers this range. The text should explicitly state that the proof for this range uses only the touching-sphere comparison in Corollary 3.3 and does not rely on the lens/rolling-ball arguments.","section":"Definition 1.1 / RΣ"}],"recommendation":"major_revision","confidential_remarks":"The paper is worth publishing after the hyperbolic λ≤1 case is completed. The missing horosphere stability computation is short and the authors likely have it; I would not reject. Please ensure the revised version extends Proposition 3.5 to all λ>0 or otherwise treats λ≤1 separately, and makes the λ-convexity preservation argument explicit."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper proves a genuinely new necessary condition for reverse isoperimetric maximizers under λ-convexity: no C² maximizers exist, and on smooth patches the smallest principal curvature must equal λ. That is a real step forward, especially since prior work identified maximizers only in low-dimensional Euclidean space. The lens-containment argument (Lemmas 3.1–3.3) is clean, and the local variation scheme is classical but well executed.\n\nThat said, the main theorem as stated overreaches. Proposition 3.5 — the workhorse — is explicitly stated only for λ∈I_Σ, where a supporting sphere of radius R_Σ(λ) exists. In hyperbolic space with λ≤1, that radius does not exist. The proof of Theorem 1.2(i) invokes Proposition 3.5 anyway. Compounding this, Proposition 3.4 supplies the needed strong stability only for CMC pieces that are not contained in a horosphere. A C² λ-convex body with λ≤1 can contain a horosphere patch, whose principal curvature is 1>λ. Nothing in the paper rules out such a patch in a maximizer, and no stability computation for horospheres is given. So the second variation step in Proposition 3.5 lacks justification precisely in the regime where it is needed for the full hyperbolic claim. The preservation of λ-convexity under the constructed variation is also asserted without a detailed proof; that one looks fixable, but it is another loose end.\n\nFor λ∈I_Σ in any of the three space forms, the proof seems coherent and the conclusion appears sound. The Euclidean and spherical parts of Theorem 1.2 should survive scrutiny. The problem is the mismatch between the theorem statement and the lemmas it relies on.\n\nThis is a paper worth refereeing seriously. The result is plausible and important; the gap is identifiable and probably repairable, either by restricting Theorem 1.2(i) to λ∈I_Σ or by extending the stability argument to horosphere patches and generalizing Proposition 3.5. I would send it to peer review and ask the authors to address hyperbolic λ≤1 explicitly.","headline":"Genuinely new variational result with a real gap in the hyperbolic λ≤1 case; the Euclidean and spherical parts look solid.","tokens_in":8612,"tokens_out":3181,"would_cite":true,"duration_ms":34053,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["52A40","53C42","53C40"],"pacs":[],"model":"deepseek-v4-flash","headline":"The reverse isoperimetric problem has no C^2 maximizers: any smooth patch of a maximizer must have its smallest principal curvature equal to λ.","keywords":["reverse isoperimetric problem","λ-convexity","perimeter maximizer","principal curvature","constant mean curvature","stability operator","space forms","variational method"],"falsifier":"Exhibit a smooth (C^2) λ-convex body in hyperbolic space with λ < 1 whose boundary has a horosphere patch and compute the second variation of area for all volume-preserving compact variations: if any such variation has non-positive second variation while the first vanishes, the improvement lemma fails for that patch and the no-smooth-maximizer claim for λ < 1 collapses. More directly, producing any smooth λ-convex body whose area is greater than every λ-convex body of the same volume would refute the theorem.","tokens_in":7773,"feed_emoji":"📐","tokens_out":8812,"duration_ms":86404,"temperature":0.7,"pith_summary":"This paper studies a 'reverse' isoperimetric question: among all convex bodies of a fixed volume whose boundary curvature is bounded below by λ, can one have the largest possible surface area? In Euclidean, spherical, and hyperbolic space, the authors prove that no maximizer can have a smooth (C^2) boundary, and that for λ in the range where a geodesic sphere of curvature λ exists, any smooth patch of a maximizer must have smallest principal curvature exactly λ. The proof works by showing that any strictly λ-convex smooth patch can be deformed in a volume-preserving way to increase surface area, contradicting maximality. This rules out smooth maximizers and forces any maximizer to develop singularities, with curvature saturating on its smooth parts.","feed_headline":"No smooth maximizers exist for the reverse isoperimetric problem","feed_subtitle":"In Euclidean, spherical, and hyperbolic space, any smooth patch of a maximizer is pinned to the curvature lower bound.","key_machinery":"The key object is the stability operator T = Δ + Ric(ν,ν) + |A|^2 acting on normal variations of a constant-mean-curvature patch, together with the sufficient condition for strong stability: existence of a positive function u with T(u) ≤ 0. In the space forms this condition is verified by exhibiting a Killing field X with ⟨ν, X⟩ > 0. Proposition 3.5 is the load-bearing variational step: it builds two-parameter deformations supported in a strictly λ-convex patch, uses the implicit function theorem to fix volume, and shows the first variation of area is positive if mean curvature varies, or the second variation is positive (via −∫ vT(v) > 0) if mean curvature is constant. This is what forces t","core_discovery":"The paper's central claim is Theorem 1.2: for any λ > 0 and any nontrivial compact λ-convex body K in Euclidean, spherical or hyperbolic space that maximizes surface area among all λ-convex bodies of the same volume, the boundary ∂K is not C^2, and—when λ lies in the parameter range for which a geodesic sphere of curvature λ exists—the smallest principal curvature on every C^2 patch of ∂K is constantly λ. The proof proceeds by contradiction: if a C^2 patch were strictly λ-convex (smallest curvature > λ), Proposition 3.5 constructs a compactly supported, volume-preserving deformation that strictly increases area, so K could not be a maximizer. A geometric lemma first shows that every nontrivi","pith_inferences":["The saturation principle suggests a free-boundary picture: maximizers should be assembled from pieces of λ-umbilic hypersurfaces (geodesic spheres of curvature λ) meeting along singular creases, analogous to the known Euclidean λ-lens examples.","One could test the mechanism numerically: running the volume-preserving area-increasing deformation on a smooth strictly λ-convex body should drive it toward a singular, curvature-saturated shape; this would offer a constructive route to maximizers in dimensions where none are explicit.","A direct stability check for horosphere caps in hyperbolic space would determine whether the no-smoothness result for λ<1 follows from the same variational mechanism or needs a separate argument.","The two-parameter variation technique may carry over to other shape optimization problems with non-open admissible classes, such as maximizing area under a prescribed lower bound on mean width."],"forward_implications":["If the theorem is correct, the reverse isoperimetric problem admits no C^2 maximizer in Euclidean, spherical, or hyperbolic space for any λ > 0; maximizers, if they exist, must be singular.","Any smooth portion of a maximizer's boundary is curvature-saturated: its smallest principal curvature is identically λ, so the boundary cannot be strictly convex at a smooth point.","The variational obstruction also applies to spherical and hyperbolic settings, where no explicit maximizers were previously known, so the result substantially narrows the search.","The proof shows that a volume-preserving deformation increasing area exists whenever a strictly λ-convex smooth patch is present, making such patches structurally unstable as maximizers."],"fun_headline_variants":["No C^2 maximizers exist for reverse isoperimetric problem","Smooth maximizers ruled out in Euclidean, spherical, hyperbolic","Maximizer boundary never smooth: curvature hits lower bound","Curvature pinned to λ on smooth patches of area maximizers"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The proof's local improvement step needs every constant-mean-curvature smooth patch of a maximizer's boundary to be strictly stable under volume-preserving perturbations; in hyperbolic space this is established only for patches that are not horospheres, so for λ ≤ 1 the claim outruns the stability argument as written.","fun_headline_variants_meta":{"raw":{"variants":["No C^2 maximizers exist for reverse isoperimetric problem","Smooth maximizers ruled out in Euclidean, spherical, hyperbolic","Maximizer boundary never smooth: curvature hits lower bound","Curvature pinned to λ on smooth patches of area maximizers"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000805,"raw_usage":{"total_tokens":3331,"prompt_tokens":661,"completion_tokens":2670,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":405,"completion_tokens_details":{"reasoning_tokens":2598}},"tokens_in":405,"tokens_out":2670,"duration_ms":18205,"temperature":1.0,"reasoning_tokens":2598,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-04T00:06:40.204909+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Exhibit a smooth (C^2) λ-convex body in hyperbolic space with λ < 1 whose boundary has a horosphere patch and compute the second variation of area for all volume-preserving compact variations: if any such variation has non-positive second variation while the first vanishes, the improvement lemma fails for that patch and the no-smooth-maximizer claim for λ < 1 collapses. More directly, producing any smooth λ-convex body whose area is greater than every λ-convex body of the same volume would refute the theorem.","supporting_citations":[],"review_version":1}