{"id":"df817cba-56de-4e47-a0e4-049c44ccbccd","arxiv_id":"2510.01647","paper_version":4,"verdict":"REJECT","confidence":"HIGH","novelty_score":3.0,"correctness_risk":"high","formal_verification":"none","parameter_count":0,"one_line_summary":"The paper's main isoperimetric and Sobolev claims rest on a weight assumption that is impossible as written, so the outside-convex-set theorems are vacuous.","lead":"This paper tries to prove sharp geometric inequalities—weighted isoperimetric and Sobolev—for droplets outside convex containers. The catch is that the weight condition it relies on cannot be satisfied by any positive function, so the main results are empty as stated.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Condition (1.5) is empty: no positive α-homogeneous weight on R^n can have a concave α-th root, so Theorems 1.1, 1.2, and 1.4 are vacuous as stated.","rationale":"The reader's weakest assumption is exactly the load-bearing failure. The paper's central claim is a universal weighted capillary isoperimetric inequality, but its stated hypothesis (1.5) describes an empty class of weights. This is not a matter of outside-consensus disagreement; it is an internal inconsistency derivable from the paper's own Lemma 3.5. The ABP argument, the capillary Schwarz symmetrization, and the Sobolev application may be coherent conditional steps, but conditional theorems with empty hypotheses prove nothing. Even if one charitably reinterpreted (1.5) as holding only on a cone, the theorem statements would need explicit revision, and the 'outside convex sets' claim would still depend on the λ_w-ABP property, which is assumed for arbitrary convex E and verified only for the half-space; Remark 1.1 itself states the general convex case is open. Thus the rejection is warranted and the verdict should remain unchanged. No ad hominem is intended; the issue is purely mathematical and located in the paper's own equations.","tokens_in":35218,"tokens_out":3776,"duration_ms":33550,"concrete_test":"Set y = -x in Lemma 3.5's inequality (3.27). Since w is α-homogeneous, ∇w(x)·x = α w(x), so the inequality becomes α(w(-x)/w(x))^{1/α} ≤ -α. The left-hand side is strictly positive because w(-x) > 0. This one-line algebraic check, using only Euler's homogeneous function theorem and the paper's own Lemma 3.5, settles that no function satisfying (1.5) exists. Alternatively, test the simplest candidate w(x)=|x|^α: its α-th root |x| is not concave on R^n, and no positive homogeneous modification can avoid the y=-x contradiction.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central hypothesis (1.5) is internally inconsistent. Let v = w^{1/α}. Then v > 0 and v is 1-homogeneous. Concavity of v on R^n applied at x and -x gives v(0) ≥ (v(x)+v(-x))/2. Since v(0)=0 by homogeneity, this forces v(x)+v(-x) ≤ 0, contradicting v > 0. The same contradiction appears inside the paper: Lemma 3.5 states that concavity of w^{1/α} is equivalent to α(w(y)/w(x))^{1/α} ≤ ∇w(x)·y/w(x) for all x,y ∈ R^n. Taking y = -x and using Euler's identity ∇w(x)·x = α w(x) yields α(w(-x)/w(x))^{1/α} ≤ -α, impossible because the left side is positive. Thus the set of admissible weights satisfying (1.5) is empty. No evenness assumption is needed for this contradiction; it follows from positivity, homogeneity, and concavity alone. Consequently Theorem 1.1, Theorem 1.2, and Theorem 1.4 have no instantiation: there is no positive weight for which the claimed weighted capillary isoperimetric inequality, Pólya-Szegő principle, or sharp Sobolev inequality is asserted. If the authors intended to restrict (1.5) to a cone, the statements would need to be revised, and even then the λ_w-ABP property is assumed for arbitrary convex E but verified only for the half-space; Remark 1.1 concedes the general convex case is open.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript studies weighted capillary isoperimetric inequalities outside a closed convex set E. It assumes a positive, even, α-homogeneous weight w with w^{1/α} concave on R^n, and introduces a λ_w-ABP property for E. Under these hypotheses it claims a sharp weighted capillary isoperimetric inequality comparing all sets with given weighted volume to the spherical cap B^λ outside the half-space, and derives a Pólya-Szegő principle and a sharp weighted Sobolev inequality. The technical core is an ABP argument in Section 3 leading to inequality (3.33), followed by an approximation argument in Section 4 and symmetrization applications in Section 5. The paper also imports the half-space sharp Sobolev inequality from [19] as an external benchmark. However, the central weight condition (1.5) is internally inconsistent: no positive 1-homogeneous concave function v = w^{1/α} exists on all of R^n. This is visible inside the paper at Lemma 3.5, Eq. (3.27), where setting y = -x gives a positive quantity bounded above by -α. Theorems 1.1, 1.2 and 1.4 are therefore vacuous as stated.","tokens_in":35554,"tokens_out":5982,"duration_ms":50021,"significance":"If corrected, the ABP chain (3.33) and the use of [19] would form a plausible route to sharp capillary Sobolev inequalities, and the equality analysis in Proposition 3.2 is a useful contribution. The derivation is not circular in the fitted-parameter sense: no parameter is fitted and the half-space Sobolev constant is imported as an external benchmark. But the empty weight class and the unverified λ_w-ABP property for general convex E mean that, as written, the paper provides no instance of its main theorems. This is a foundational obstruction, not a presentation issue.","major_comments":[{"comment":"The admissible weight class is empty. Let v = w^{1/α}. Then v is positive, 1-homogeneous and concave on R^n. Concavity gives v(0) ≥ (v(x)+v(-x))/2; since v(0)=0 and v(x), v(-x)>0, this is impossible. Equivalently, taking y = -x in (3.27) and using Euler's identity ∇w(x)·x = αw(x) yields α(w(-x)/w(x))^{1/α} ≤ -α, impossible. Evenness, assumed in the abstract, does not remove the contradiction. Therefore Theorems 1.1, 1.2 and 1.4 quantify over no admissible weight and are vacuous as stated.","section":"§1, Eq. (1.5); §3, Lemma 3.5, Eq. (3.27)"},{"comment":"The λ_w-ABP property is assumed for arbitrary closed convex E, but the only verification provided is for the half-space (end of §4). Remark 1.1 concedes that the general convex case is open. Thus the paper's headline claim of an isoperimetric inequality 'outside convex sets' is conditional on a hypothesis whose scope is not established. Even after repairing (1.5) by restricting the concavity to a cone, this gap would remain for Theorem 1.1.","section":"§1, Theorem 1.1; Definition 3.2/(3.23)"},{"comment":"The reduction to smooth data via Lemma 4.3 is incomplete: Proposition 3.2 requires the approximating convex sets E_h to satisfy the λ_w-ABP property, but Lemma 4.3 does not establish that this property is preserved under Kuratowski convergence or under the chosen approximation. Without such a preservation statement, the passage from E to E_h is not justified for general convex sets.","section":"§4, proof of Theorem 1.1"}],"minor_comments":[{"comment":"The proof says 'concave outside E' while the statement and condition (1.5) say concave on R^n. This inconsistency is not merely cosmetic; it is connected to the vacuity of the weight class.","section":"§3, Lemma 3.5"},{"comment":"The statements say 'Ω ⊂ E is a set of finite perimeter', but the definitions and boundary conditions require Ω ⊂ R^n \\ E (or E^c). This appears to be a typo, but it makes the theorems hard to read.","section":"§1, Theorems 1.3 and 1.4"},{"comment":"'Kuratoswki' should be 'Kuratowski'.","section":"§4, Definition 4.2"},{"comment":"There are numerous typographical and reference errors (e.g., 'D ´laz', 'Poincur´ e', inconsistent formatting of B^λ and B^λ_r). A careful editorial pass is needed.","section":"Throughout"}],"recommendation":"reject","confidential_remarks":"The rejection is based on the internal contradiction in condition (1.5), which is exposed by the paper's own Lemma 3.5. If the authors intend cone-restricted weights, the statements and proofs must be reformulated, and the λ_w-ABP property needs nontrivial examples beyond half-spaces before an 'outside convex sets' claim is supportable. This is a substantial revision, not a local fix."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague, before you spend time on this, know the main theorems are vacuous as stated. Condition (1.5) asks for a positive, even, α-homogeneous weight w with w^{1/α} concave on all of R^n. That class is empty. The paper's own Lemma 3.5 characterizes concavity via (3.27); setting y = -x and using Euler's identity gives α(w(-x)/w(x))^{1/α} ≤ -α, impossible for positive w. Equivalently, no positive one-homogeneous concave function exists on R^n. So Theorems 1.1, 1.2, and 1.4 quantify over no admissible weight. This is not a subtle gap; it is in the paper's own equations.\n\nWhat is genuinely there: the conditional framework is coherent. The weighted capillary gauge, the λ_w-ABP property, the reformulation of J_{w,λ} as an anisotropic weighted perimeter, and the symmetrization argument are all reasonable steps. The proof of Proposition 3.2 is a careful adaptation of the ABP method, and the subcritical approximation in Theorem 1.4 is a plausible route to the Sobolev inequality. The authors engage honestly with Cabré–Ros-Oton–Serra, Ciraolo–Figalli–Roncoroni, and the recent Fusco–Julin–Morini–Pratelli work. There is no fitted parameter and the half-space constant is imported from [19], so the derivation is not circular in the prediction sense.\n\nThe soft spots beyond the emptiness: the 'outside convex sets' part depends on the λ_w-ABP property, verified only for half-spaces; Remark 1.1 concedes the general case is open. So even if (1.5) were replaced by a cone condition, Theorem 1.1 would be conditional. There are also minor technical gaps in the limit interchanges for the Sobolev constant, but they are not the issue. The central problem is that no single example of an admissible weight exists.\n\nBottom line: this deserves a desk reject in current form. The fix is not hard in principle—restrict (1.5) to a convex cone and prove λ_w-ABP for a wider class, or drop the outside-convex-set claim—but as written the statement is vacuous. I would not cite it and would not send it to a referee until the hypothesis is corrected. If the authors rework it, the symmetrization part might be worth a second look.","headline":"The paper's central weight assumption (1.5) is empty—Lemma 3.5 forces a contradiction—so the headline theorems are vacuous despite a coherent conditional ABP/symmetrization framework.","tokens_in":36126,"tokens_out":3248,"would_cite":false,"duration_ms":27374,"reading_group":"maybe","serious_thinker":"no","would_accept_peer_review":false},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35A23","35J92","49Q20","52A40"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper claims that, for a class of homogeneous weights, spherical caps are the universal minimizers of weighted capillary energy outside any closed convex set — a result that would yield sharp weighted Sobolev inequalities beyond the hal","keywords":["capillary isoperimetric inequality","weighted Sobolev inequality","ABP method","homogeneous weights","convex sets","capillary Schwarz symmetrization","Pólya-Szegő principle"],"falsifier":"A direct calculation with $y=-x$ in the paper's inequality (3.27) gives $\\alpha \\left( \\frac{w(-x)}{w(x)} \\right)^{1/\\alpha} \\leq -\\alpha$, contradicting positivity of $w$; this shows the main theorem is vacuous as stated. A meaningful test would be to exhibit a weight satisfying the assumptions on a cone and then verify the $\\lambda_w$-ABP property (3.19) for a non-flat convex $E$, or to find a counterexample if no such $E$ satisfies it.","tokens_in":34995,"feed_emoji":"📐","tokens_out":8553,"duration_ms":65254,"temperature":0.7,"texified_at":"2026-08-05T20:33:10.495169+00:00","pith_summary":"The paper aims to prove that outside any closed convex set, the weighted capillary energy of a region of fixed weighted volume is minimized by a spherical cap resting on a flat part of the boundary. The intended proof adapts the Alexandrov-Bakelman-Pucci method to a weighted Neumann problem and introduces an anisotropic gauge that recasts the capillary energy as a weighted perimeter. If the main theorem held, it would yield a sharp weighted Sobolev inequality outside convex sets with the same best constant as in the half-space, plus a weighted Pólya-Szegő principle via a new capillary Schwarz symmetrization. The paper also identifies the equality case: minimizers are isometric to the spherical cap on a flat boundary. However, the weight class (1.5) as stated — positive, even, $\\alpha$-homogeneous with concave $\\alpha$-th root on all of $\\mathbb{R}^n$ — is empty; concavity forces $w(-x) \\leq -w(x)$, contradicting positivity, so the theorem is vacuous unless the assumption is restricted to a cone.","texify_model":"deepseek-v4-flash","texify_usage":{"total_tokens":11184,"prompt_tokens":881,"completion_tokens":10303,"prompt_tokens_details":{"cached_tokens":0},"prompt_cache_hit_tokens":0,"prompt_cache_miss_tokens":881,"completion_tokens_details":{"reasoning_tokens":9399}},"feed_headline":"Spherical cap minimizes weighted capillary energy outside convex sets","feed_subtitle":"A new ABP proof aims to extend the half-space result, but the stated weights cannot exist on R^n.","key_machinery":"The $\\lambda_w$-ABP property (Definition 3.2): for any finite set $K \\subset \\partial E$ and any function $v: K \\to \\mathbb{R}$, the weighted measure of $B^\\lambda$ is bounded by the weighted measure of $B^\\lambda_v \\cap B_1$, where $B^\\lambda_v$ is the union of subdifferentials of $v$ whose normal component exceeds $\\lambda$. This property converts the ABP contact argument into a purely geometric comparison on the boundary of $E$. A second key tool is the capillary gauge $\\tilde{F}_{\\lambda,w}(\\xi) = |\\xi| + \\nabla h \\cdot \\xi$, where $h$ solves a degenerate Neumann problem; this gauge turns the weighted capillary energy into a weighted anisotropic perimeter. The concavity condition on $w^{1/\\alpha}$ supplies an algebraic inequality (3.31) that replaces the arithmetic-geometric mean step in the classical ABP proof.","core_discovery":"Theorem 1.1 asserts that for a closed convex set $E$ satisfying the $\\lambda_w$-ABP property (3.23) and any weight $w$ satisfying (1.5), every finite-perimeter $\\Omega$ outside $E$ with weighted volume equal to that of $B^\\lambda$ satisfies $J_{w,\\lambda}(\\Omega;\\mathbb{R}^n\\setminus E) \\geq J_{w,\\lambda}(B^\\lambda;\\mathbb{R}^n\\setminus H)$, with equality exactly when $\\Omega$ is isometric to $B^\\lambda$ on a flat part of $\\partial E$. Theorem 1.2 proves the same for the half-space without the ABP assumption, and Theorem 1.4 derives the sharp weighted capillary Sobolev inequality outside convex sets. The proof route: solve a weighted Neumann problem, show its solution is a viscosity supersolution, use the $\\lambda_w$-ABP measure comparison on $\\partial E$ to control the subdifferential image, and then convert the energy via\n","pith_inferences":["Since the stated weight class (1.5) is empty on R^n, the paper's natural reading is as a cone or orthant statement; on such a domain the concavity condition is compatible with homogeneity, and the ABP chain (3.33) is plausibly sound.","The λ_w-ABP property is verified only for the half-space in the paper; checking it for a genuinely curved convex boundary (e.g. a cylinder or a ball) would be a direct test of whether the general convex-set claim is meaningful.","The gauge reformulation (2.10) relies on a solution h to the degenerate Neumann problem (2.9); for non-smooth convex sets the existence and regularity of h is not addressed, so the weighted anisotropic perimeter representation may require a limiting argument.","If the weight condition is repaired on a cone, the critical Sobolev exponent p*_α = (n+α)p/(n+α-p) and the extremal functions (5.16) would need re-derivation, since the cone geometry changes the homogeneity constant."],"forward_implications":["If Theorem 1.1 is correct, the spherical cap is the unique (up to isometry) minimizer of weighted capillary energy outside every convex set satisfying the λ_w-ABP property.","The sharp weighted capillary Sobolev inequality (Theorem 1.4) would hold for all convex obstacles in the class, with the same best constant as the half-space result.","The capillary Schwarz symmetrization of Theorem 1.3 would give a rearrangement tool that preserves weighted L^p norms and decreases the weighted anisotropic gradient energy outside convex domains.","Equality cases would be rigid: only flat-boundary spherical caps achieve equality in the isoperimetric inequality."],"fun_headline_variants":["Weighted capillary isoperimetry extends to convex exteriors","Sharp Sobolev inequality outside convex sets via ABP","Capillary Schwarz symmetrization yields weighted Sobolev bound","New isoperimetric and Sobolev inequalities for convex complements","Convex set exterior: weighted capillary and Sobolev inequalities"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The paper assumes a positive, even, $\\alpha$-homogeneous weight $w$ on $\\mathbb{R}^n$ with concave $\\alpha$-th root; this class is empty, because concavity of the $1$-homogeneous root forces $w(-x) \\leq -w(x)$ for every $x$, contradicting positivity.","fun_headline_variants_meta":{"raw":{"variants":["Weighted capillary isoperimetry extends to convex exteriors","Sharp Sobolev inequality outside convex sets via ABP","Capillary Schwarz symmetrization yields weighted Sobolev bound","New isoperimetric and Sobolev inequalities for convex complements","Convex set exterior: weighted capillary and Sobolev inequalities"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000697,"raw_usage":{"total_tokens":2962,"prompt_tokens":698,"completion_tokens":2264,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":442,"completion_tokens_details":{"reasoning_tokens":2180}},"tokens_in":442,"tokens_out":2264,"duration_ms":15830,"temperature":1.0,"reasoning_tokens":2180,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-04T12:52:13.623962+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A direct calculation with $y=-x$ in the paper's inequality (3.27) gives $\\alpha \\left( \\frac{w(-x)}{w(x)} \\right)^{1/\\alpha} \\leq -\\alpha$, contradicting positivity of $w$; this shows the main theorem is vacuous as stated. A meaningful test would be to exhibit a weight satisfying the assumptions on a cone and then verify the $\\lambda_w$-ABP property (3.19) for a non-flat convex $E$, or to find a counterexample if no such $E$ satisfies it.","supporting_citations":[],"review_version":1}