{"id":"b7ae6e22-c140-4280-b76b-7bb50699d216","arxiv_id":"2509.20257","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"The capillary Blaschke-Santalo inequality holds for unconditional strictly convex capillary hypersurfaces with theta in (0, pi/2); for theta in (pi/2, pi) the volume product is unbounded.","lead":"This paper proves a conjectured volume inequality for curved caps meeting a flat wall at a fixed acute angle, assuming the cap is symmetric in all coordinate directions. It also shows the analogous volume product is unbounded for obtuse contact angles, so the conjecture cannot extend there.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Constant in Theorem 1.1/(2.3) is off by a factor of n: the proof yields vol(K)/(2n), not vol(K)/2^n, so the stated capillary implication and equality case fail.","rationale":"The reader's weakest assumption was the geometric identification sΣ(ζ)=h_K(ζ+cosθE_n) for K=bΣ∪R(bΣ). That step is actually sound: for x=ζ+cosθE_n in the cap, h_K(x)=h_{bΣ}(x)=<X(ζ),x>=sΣ(ζ), so the bridge to the analytic inequality is not the main risk. My read found a more substantive algebraic error in the main theorem: the derivation leading to display (2.3) gives a denominator 2n, not 2^n. With the stated 2^n, the 'In particular' capillary inequality does not follow, and the equality characterization fails for n≥3. The error is almost certainly a typo and the proof strategy appears viable once the constant is corrected, so the verdict remains CONDITIONAL rather than ACCEPT; the authors should correct (2.3) and Theorem 1.1 and recheck the equality argument. The reader did notice the constant in (2.3) was misstated but treated it as minor; my analysis shows it is load-bearing, hence partial agreement.","tokens_in":9556,"tokens_out":19273,"duration_ms":134445,"concrete_test":"Independently re-derive the passage from (2.2) to (2.3): insert the three identities A=(c/2^{n-1})I, B=(nc/2^n)vol(K), C=(nc/2^{n-1})vol(bCθ) into A B ≤ C² and simplify. If the result is vol(K)/(2n)I ≤ vol(bCθ)², then the printed constant 2^n is wrong. Additionally test sharpness with n=3, K=B^3, θ=π/2 (or the θ→π/2 limit), where equality must hold for the 2n constant and fails for 2^n.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"Combining (2.2) with the three displayed identities before (2.3) gives, with I = ∫_{S^{n-1}_θ} h_C^{n+1}/h_K^n dσ, A = (c/2^{n-1})I, B = (nc/2^n)vol(K), C = (nc/2^{n-1})vol(bCθ). Then A B ≤ C² simplifies to vol(K)/(2n) I ≤ vol(bCθ)², not vol(K)/2^n I ≤ vol(bCθ)². This is not cosmetic: substituting K = bΣ ∪ R(bΣ) and using vol(cΣ*) = (1/n)I, the printed denominator 2^n yields only (2n/2^n) vol(bΣ)vol(cΣ*) ≤ vol(bCθ)², which for n ≥ 3 is strictly weaker and does not give the claimed capillary inequality. The equality statement is also inconsistent as stated: for Σ = bCθ (so K = C), the correct 2n inequality is sharp, whereas the 2^n inequality is strict (e.g. n=3, θ→π/2, K=B^3 gives LHS = π²/3, RHS = 4π²/9). The theorem as printed therefore needs its constant corrected to 2n, and the equality-case passage re-verified under that correction.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper addresses the conjectured capillary Blaschke–Santaló inequality for even, strictly convex capillary hypersurfaces in the upper half-space. For θ∈(0,π/2), the authors claim the inequality holds for all unconditional such hypersurfaces, with equality if and only if the hypersurface is a dilate of the capillary spherical cap Cθ. The proof strategy is functional-analytic: they introduce the convex body C = bCθ ∪ R(bCθ), prove concavity of V(√x) for V=(1/2)p_C², import a gradient inequality from [CKLR24], apply the Prékopa–Leindler inequality, and reduce the desired capillary inequality to a weighted integral inequality for unconditional convex bodies K. The paper also sketches two examples showing that for θ∈(π/2,π) the capillary volume product is unbounded, and it derives a linearized spectral inequality.","tokens_in":9911,"tokens_out":28916,"duration_ms":196411,"significance":"If the constant issue described below is corrected, this is a significant and clean result: it proves the conjectured capillary Blaschke–Santaló inequality in the unconditional class, with a sharp equality statement. The proof is genuinely non-circular: it imports the capillary polar-volume formulas from [MWW25] and a gradient inequality from [CKLR24], uses external results (Prékopa–Leindler and Dubuc's equality case) but introduces no free parameters or fitting. The unboundedness for obtuse contact angles is also an interesting complement. The main theorem is of clear interest to convex geometry and capillary hypersurface theory.","major_comments":[{"comment":"The displayed simplification leading to (2.3) has the wrong constant and the equality statement is inconsistent with it. With I = ∫_{Sθ} h_C^{n+1}/h_K^n dσ, the three preceding identities give A = (nc/2^n)V(K), B = (c/2^{n-1})I, C = (nc/2^{n-1})V(bCθ). Substituting AB ≤ C² yields V(K)/(2n) I ≤ V(bCθ)², not V(K)/2^n I ≤ V(bCθ)². The printed 2^n inequality is weaker; it does not imply the capillary inequality, and its equality case is false (for n≥3, K=C gives strict inequality). Please correct (2.3) and Theorem 1.1 to the 2n form and re-verify the equality-case argument under the corrected statement.","section":"§2.1, Eq. (2.3) and Theorem 1.1"},{"comment":"The proof uses the identification h_K(ζ+cosθE_n)=sΣ(ζ) for K=bΣ∪R(bΣ) to convert the capillary polar volume into the integral over Sθ. This is not proved or referenced. It is essential: without it the geometric inequality for Σ does not follow from the analytic inequality for K. Please supply a short supporting-hyperplane argument (the point on Σ with Euclidean unit normal u=ζ+cosθE_n supports bΣ, hence K, in direction u) or a precise reference.","section":"§2.1, passage from Σ to K"},{"comment":"The two examples are constructed in R². The abstract claims no finite upper bound, presumably in every dimension, but the text only says the planar bodies may be rotated about the x_n-axis. To make the claim rigorous for general n, the rotation reduction and the corresponding estimates for the n-dimensional body and its capillary polar need to be stated explicitly; otherwise the result is proved only for n=2.","section":"§3, unboundedness"}],"minor_comments":[{"comment":"In the first integral after 'By Lemma 2.2 and Lemma 2.3', the integrand should be e^{-1/2 h_K²(y)} (equivalently e^{-1/2 p_{K°}²(y)}), not e^{-1/2 p_K²(y)}. Since Φ*=1/2 h_K², the printed p_K is a typo that would make the subsequent h_K formula unjustified.","section":"§2.1, display before spherical coordinates"},{"comment":"In the equality case, from p_K(e^t)=p_C(e^{t+w/2}) the correct diagonal matrix is diag(e^{-w_i/2}), not diag(e^{w_i/2}), to obtain p_{AK}=p_C. The final conclusion is unaffected, but the sign should be fixed.","section":"§2.1, equality-case paragraph"},{"comment":"The notation ΔCθ is introduced only via the phrase 'centro-affine Laplacian' and a reference; a one-line definition or formula would improve readability.","section":"§2.2, Theorem 1.2"},{"comment":"In the construction of Kλ, the displayed inequality for s_{Σλ}(ζ) is terse; a short explanation of why the other parts of the union do not affect the capillary support function on Dθ would be helpful.","section":"§3, Example 1"}],"recommendation":"major_revision","confidential_remarks":"The core mathematical idea is sound and the capillary inequality is likely correct after the constant correction. The present version, however, states a weaker analytic inequality and an equality case that cannot hold for that statement; this suggests the final simplification was not checked carefully. The unboundedness section also needs a clearer higher-dimensional reduction. I recommend major revision rather than rejection; the fixes are local and the main theorem is defensible."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"You should know two things. First, the paper actually proves the conjecture it claims in the unconditional class, and the obtuse-angle unboundedness result is new. Second, Theorem 1.1 as printed has a wrong denominator: the proof gives 2n, not 2^n, and the stated 'in particular' and equality case only follow after that correction.\n\nWhat is genuinely new: the unconditional capillary Blaschke–Santaló inequality, the equality characterization, and the unboundedness for θ > π/2. The argument is clean: build a symmetric body C from the cap, show V(√x) is concave, use a functional form of the Blaschke–Santaló inequality from CKLR24, and convert the Prékopa–Leindler step into the capillary volume product using the capillary polar volume formula from MWW25. Theorem 1.2, a linearization, is a reasonable bonus. Section 3 gives two explicit families showing the product is unbounded for obtuse angles; the construction is a bit ad hoc but it checks out.\n\nThe constant bug is real and not cosmetic. Combining (2.2) with the three identities before (2.3) gives vol(K)/(2n) I ≤ vol(bCθ)², not vol(K)/2^n I ≤ vol(bCθ)². Since I = n vol(cΣ*) and vol(K ∪ R(K)) = 2 vol(bΣ), the 2n denominator is exactly what produces vol(bΣ)vol(cΣ*) ≤ V(bCθ)². The printed 2^n leaves a factor 2n/2^n on the left, so for n ≥ 3 the capillary implication and the equality claim do not follow as written; e.g., n=3, θ→π/2, K=B³ gives strict inequality with the printed constant. This is a one-character fix in Theorem 1.1 and (2.3), and the proof already contains the right constants, so I do not treat it as fatal—but it must be corrected before publication. The reader's note about Lemma 2.2 omitting a homogeneity normalization is also worth addressing: add one line explaining the identity is for the normalized support function or state the homogeneous version explicitly. The equality-case argument is terse but credible.\n\nWho is this for: convex geometers working on capillary problems or functional Blaschke–Santaló inequalities. It deserves a serious referee; I would send it out with the expectation that the authors fix the constant and tighten the cited identity. After that, it is a solid advance.","headline":"The main theorem is right but the printed constant is wrong: it should be 2n, not 2^n; with that fix the paper is a solid proof of the unconditional capillary Blaschke–Santaló case.","tokens_in":10339,"tokens_out":11811,"would_cite":true,"duration_ms":106100,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["52A40","53A15"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves the conjectured capillary Blaschke–Santaló inequality for unconditional strictly convex capillary hypersurfaces when the contact angle is acute, and shows the volume product is unbounded for obtuse angles.","keywords":["capillary hypersurface","Blaschke–Santaló inequality","unconditional convex body","volume product","Prékopa–Leindler inequality","Legendre transform","centro-affine","contact angle"],"falsifier":"Construct an unconditional, strictly convex capillary hypersurface with θ ∈ (0, π/2) (e.g., a small diagonal perturbation of the spherical cap) and compute vol(bΣ) vol(cΣ*) numerically; if any such product exceeds vol(bC_θ)², the central claim collapses. Alternatively, verify the paper's obtuse-angle example by evaluating the support-function bound h_{bΣ}(ψ) for the ellipse family and checking that the integral grows linearly in b.","tokens_in":9469,"feed_emoji":"📐","tokens_out":2890,"duration_ms":119798,"temperature":0.7,"pith_summary":"The paper proves a half-space analog of the classical Blaschke–Santaló inequality: among convex bodies that meet the boundary plane at a fixed acute angle and are symmetric in each coordinate direction, the spherical cap maximizes the product of the body's volume and the volume of its capillary polar body. This confirms a conjecture for the unconditional case. For obtuse contact angles, the paper constructs families of convex bodies for which this volume product grows without bound, so no such inequality can hold there. If correct, the result extends a cornerstone of convex geometry to a capillary setting and provides a sharp equality characterization.","feed_headline":"Spherical cap wins capillary volume-product bound","feed_subtitle":"For acute contact angles, unconditional shapes satisfy a Blaschke–Santaló inequality; obtuse angles break it entirely.","key_machinery":"The proof reduces the geometric inequality to an analytic one via the Legendre-transform pair V = ½p_C² and V* = ½h_C², where C is the convex body obtained by reflecting the capillary cap across the boundary hyperplane. The core identity det D²V(x) · det D²V*(DV(x)) = 1, together with the concavity of V(√x) and the Prékopa–Leindler inequality, yields the integral inequality (2.3). The capillary polar volume is expressed through the capillary Gauss map and support function, bridging the analytic bound to the geometric volume product.","core_discovery":"For θ ∈ (0, π/2), the authors establish that for any unconditional convex body K in R^n, the weighted integral inequality vol(K)/2^n ∫_{S^{n-1}_θ} (1 − cosθ x_n)^{n+1} / h_K^n dx ≤ vol( bC_θ)^2 holds, where bC_θ is the capillary spherical cap. When K is the double of an unconditional, strictly convex capillary hypersurface Σ, this directly gives vol(bΣ) vol(cΣ*) ≤ vol(bC_θ)^2, with equality if and only if Σ is a homothetic copy of C_θ. In contrast, for θ ∈ (π/2, π), they exhibit two explicit one-parameter families of convex bodies showing that vol(bΣ) vol(cΣ*) → ∞.","pith_inferences":["The technique may extend to non-unconditional bodies if the symmetry assumption can be relaxed, though the Prékopa–Leindler argument relies on coordinate-wise convexity; this is left open by the paper.","The unboundedness constructions for obtuse angles suggest that for θ > π/2 the capillary polar volume can be made arbitrarily large by 'flattening' the body near the boundary plane, hinting at a possible phase transition in the geometry.","A quantitative stability version of the equality case might follow from known stability results for Prékopa–Leindler, though the paper does not address it explicitly."],"forward_implications":["The conjectured capillary Blaschke–Santaló inequality is true for all unconditional, strictly convex capillary hypersurfaces with θ ∈ (0, π/2), with spheres as the unique maximizers up to scaling.","A linearization (Theorem 1.2) yields a sharp second-order inequality involving the centro-affine Laplacian on the cap, giving a necessary condition for local maximality.","Corollary 2.5 provides entropy-type and L^p-type inequalities for unconditional convex bodies, extending the volume product bound to other functionals.","For θ ∈ (π/2, π), the absence of a finite upper bound shows the acute-angle condition is essential for any capillary Blaschke–Santaló-type statement."],"fun_headline_variants":["Acute angles keep capillary Blaschke-Santaló; obtuse break it","Capillary sphere maximizes volume product for acute angles","Capillary Blaschke-Santaló holds for acute, fails for obtuse","Spherical cap solves capillary inequality for acute angles","Unconditional convex bodies prove capillary Santaló for acute θ"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The proof assumes that for every unconditional, strictly convex capillary hypersurface, the capillary polar volume can be represented by the same integral formula as for the model cap, specifically that the capillary support function s_Σ equals the standard support function h_K at shifted arguments; if that identification fails for some admissible surface, the geometric inequality does not follow from the analytic bound.","fun_headline_variants_meta":{"raw":{"variants":["Acute angles keep capillary Blaschke-Santaló; obtuse break it","Capillary sphere maximizes volume product for acute angles","Capillary Blaschke-Santaló holds for acute, fails for obtuse","Spherical cap solves capillary inequality for acute angles","Unconditional convex bodies prove capillary Santaló for acute θ"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000299,"raw_usage":{"total_tokens":1498,"prompt_tokens":606,"completion_tokens":892,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":350,"completion_tokens_details":{"reasoning_tokens":804}},"tokens_in":350,"tokens_out":892,"duration_ms":8505,"temperature":1.0,"reasoning_tokens":804,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-04T15:16:44.977252+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Construct an unconditional, strictly convex capillary hypersurface with θ ∈ (0, π/2) (e.g., a small diagonal perturbation of the spherical cap) and compute vol(bΣ) vol(cΣ*) numerically; if any such product exceeds vol(bC_θ)², the central claim collapses. Alternatively, verify the paper's obtuse-angle example by evaluating the support-function bound h_{bΣ}(ψ) for the ellipse family and checking that the integral grows linearly in b.","supporting_citations":[],"review_version":1}