{"id":"9cb1addf-dd2b-4643-9436-e769b09ec0cf","arxiv_id":"2509.10859","paper_version":1,"verdict":"REJECT","confidence":"HIGH","novelty_score":5.0,"correctness_risk":"high","formal_verification":"none","parameter_count":0,"one_line_summary":"The capillary Orlicz-Minkowski problem is formulated, but the main existence theorem is unsupported because the initial solution of the continuity method is not admissible under the paper's normalization.","lead":"This paper introduces a capillary version of the Orlicz-Minkowski problem and claims existence of volume-normalized smooth even solutions via a continuity method. The proof, as written, breaks because the spherical cap, the starting point of the continuity path, fails the paper's own orthogonality condition.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Continuity method has no proven starting point: the only t=0 solution h=ℓ violates the orthogonality condition that defines H, so nonemptiness of I is not established.","rationale":"The reader's rejection hinges on a precise, checkable failure in the continuity argument. I re-derived the membership of ℓ in H: at t=0 the equation forces det(ℓij+ℓδij)=1, and the gauge coefficient is the constant a=φ'(1)/φ(1)-n-1. The two branches both exclude ℓ. This is not a matter of disagreement with a conjecture; it is a gap in the internal logic of Theorem 1.1's proof. Openness and closeness of I only matter if I is nonempty, and the paper provides no t=0 solution in H. I do not claim I must be empty for all possible h, since other solutions of the t=0 equation might conceivably lie in H; but none is exhibited, so the base case is unproved. The other issues noted by the reader—the §4 estimates apply to the normalized equation while (5.1) is unnormalized, and the Lp remark that v=0 in (1.3) is false for p≠n+1—are real but secondary. The Section 3 inequalities appear to be independent contributions and could survive a repaired existence proof, which is why the paper as a proof of Theorem 1.1 should still be rejected, without rejecting all content. No change to the reader's verdict is needed.","tokens_in":19292,"tokens_out":11231,"duration_ms":98121,"concrete_test":"Substitute h=ℓ into the definition of H in Section 5 at t=0 and verify membership. In the generic case φ'(1)≠(n+1)φ(1), choose v=ℓ-(∫_{Cθ}ℓ dξ)/(∫_{Cθ}1 dξ); then ∫_{Cθ}(φ'(1)/φ(1)-n-1)v det(ℓij+ℓδij)dξ=0, but ∫_{Cθ}ℓv dξ=∫ℓ²-(∫ℓ)²/|Cθ|>0 since ℓ is nonconstant, so ℓ∉H. In the special case φ'(1)=(n+1)φ(1), the premise holds for v=ℓ, while ∫ℓ²>0. This check shows the stated starting solution is outside H. If the authors believe I(0) is nevertheless nonempty, the test is to exhibit any positive even solution of (5.1) at t=0 lying in H, or to modify the gauge so that ℓ∈H.","verdict_should_be":"UNCHANGED","load_bearing_attack":"In the proof of Theorem 1.1, Section 5 defines H by the gauge condition that h is orthogonal to every v satisfying ∫_{Cθ}(ℓφ'(ℓ/h)/(hφ(ℓ/h))-n-1)v det(hij+hδij)=0, and defines I as the t-set for which (5.1) has a positive even solution in H. At t=0 the only solution exhibited is h=ℓ. For h=ℓ, equation (5.1) gives det(ℓij+ℓδij)=1 and the gauge coefficient reduces to the constant a=φ'(1)/φ(1)-n-1. If a≠0, the premise in H's definition is exactly that v has zero mean; taking v=ℓ-c with c=(∫_{Cθ}ℓ)/(∫_{Cθ}1) gives ∫v=0 but ∫ℓv=∫ℓ²-c∫ℓ=Var(ℓ)>0 because ℓ is nonconstant. If a=0, the premise holds for every v, in particular v=ℓ, but ∫ℓ²>0. In both branches ℓ∉H. Consequently the continuity set I is not shown to contain t=0; the statement 'h=ℓ is a solution at t=0' does not supply a point in H from which the implicit-function/openness step can start. No alternative starting solution is constructed or cited. Since nonemptiness is the base case of the continuity method, the existence proof has no proven starting point. A separate normalization mismatch between the §4 estimates (for the normalized equation (4.1)) and the unnormalized continuity equation (5.1) would also need repair, but the missing starting point is the immediate obstruction.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces a Robin-boundary analogue of the Orlicz-Minkowski problem, called the capillary Orlicz-Minkowski problem. For a convex function φ in a class O and a positive even function f on a spherical cap Cθ, the problem asks for a symmetric capillary convex body of volume one whose support function h solves φ(ℓ/h)h det(hij+hδij)=f in Cθ with Robin condition ∇μh=cotθ h on ∂Cθ. The authors define a capillary Orlicz surface area measure, establish capillary Orlicz-Minkowski and Orlicz-Brunn-Minkowski inequalities using the Alexandrov-Fenchel inequality from [48], derive a priori estimates for a normalized version of the equation, and then prove the main existence theorem by a continuity method, with a uniqueness statement when equality holds in the integral condition.","tokens_in":19623,"tokens_out":11132,"duration_ms":100956,"significance":"If the main theorem were correct, the paper would provide a meaningful extension of the Orlicz-Minkowski problem to capillary convex bodies and would unify several recent capillary Lp results. The geometric inequalities in Section 3 appear to follow from known tools and are a plausible contribution: the Orlicz-Minkowski inequality (3.3) and Orlicz-Brunn-Minkowski inequality (3.4) are derived cleanly from the Alexandrov-Fenchel inequality of [48] and Jensen's inequality. However, the existence proof has a missing base point in the continuity argument, a mismatch between the normalized estimates and the unnormalized continuity path, and an incomplete Fredholm/surjectivity step. These are load-bearing defects, so the central existence theorem is not established in the present manuscript.","major_comments":[{"comment":"The starting solution h=ℓ is not an element of the set H used in the continuity method. For h=ℓ we have det(ℓij+ℓδij)=1 and ℓ/h=1, so the coefficient in the defining condition of H reduces to the constant a=φ'(1)/φ(1)−n−1. If a≠0, the premise in H's definition is exactly ∫v=0, but the even function v=ℓ−c with c=(∫ℓ)/(∫1) has zero mean while ∫ℓv=∫ℓ²−c∫ℓ=Var(ℓ)>0 because ℓ is nonconstant. If a=0, the premise holds for every v, in particular v=ℓ, and ∫ℓ²>0. Thus ℓ∉H in either branch. The statement 'It is easy to see that h=ℓ is a solution to (5.1) for t=0' only checks the PDE, not membership in H. No alternative solution at t=0 is constructed. Consequently I is not shown to contain t=0, and the continuity method has no proven base point.","section":"Section 5, definition of H and proof of Theorem 1.1"},{"comment":"The a priori estimates of Section 4 are proved for the normalized equation (4.1), which contains the factor 1/|bΣ|; for example, Lemma 4.1 uses this factor to obtain the bound ∫ φ(ℓ/h) f ≤ n+1. The continuity equation (5.1) is not normalized. A solution h of (5.1) does not satisfy (4.1) with the same right-hand side f_t, and no rescaling argument is given. For general φ∈O, replacing h by |bΣ|^{-1/(n+1)}h does not preserve the term φ(ℓ/h) because φ is not homogeneous. The proof of closedness of I invokes the estimates (4.27), but those estimates do not apply to solutions of (5.1) as written. This is a second load-bearing gap in the existence proof.","section":"Section 4 versus Section 5"},{"comment":"The proof that Lh is surjective on H is incomplete. Lemma 5.2 shows only that every v∈H∩Ker(Lh) satisfies the integral condition ∫(ℓφ'(ℓ/h)/(hφ(ℓ/h))−n−1)v det(hij+hδij)=0. The displayed chain in the proof of Theorem 1.1 then identifies (Ker(Lh))⊥ with the orthogonal complement of the set of v satisfying that integral condition. This identification requires the reverse inclusion, which is not proved and is not evident. Moreover, H itself is defined using h in the gauge condition, so H is not a linear subspace of C^{4,α}; arguments using 'Range(Lh)=H' and orthogonal complements in H therefore lack a clear functional-analytic setting. The implicit function theorem step is not justified on the basis of the lemmas stated.","section":"Section 5, surjectivity of Lh"}],"minor_comments":[{"comment":"The orthogonality condition in Definition 1.2 is stated with respect to a function f and involves f/(hφ(ℓ/h)), while the set H in Section 5 is defined with det(hij+hδij) and no f. The equivalence between these two formulations for solutions of (5.1) should be stated explicitly, especially because f_t changes along the continuity path.","section":"Definition 1.2 and Section 5"},{"comment":"The proof of Theorem 3.3 cites 'Proposition 3.3', but no Proposition 3.3 appears in the paper; the reference should be to the appropriate definition, lemma, or variational formula.","section":"Section 3, proof of Theorem 3.3"},{"comment":"There are minor typographical issues, including 'theroy' in the Section 3 heading, a duplicated reference number [62] in the introduction, and the abstract phrase 'which can change our solutions to a spherical cap', which is unclear and should be rephrased.","section":"Throughout"}],"recommendation":"reject","confidential_remarks":"The paper relies heavily on a set of very recent preprints and on [44,48], and the novel analytic part is the continuity argument. The failure of the t=0 solution to lie in H is not a small technicality; it is a structural mismatch between the gauge condition and the only known starting point. A repair would require either a different gauge or a different starting family, and the normalization mismatch and the surjectivity gap would also need to be addressed. I do not see a way to fix these within the scope of a revision."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Quick take: the capillary Orlicz-Minkowski problem is a natural synthesis, and the two inequalities in Section 3 are genuinely new and look correct. But the continuity argument for Theorem 1.1 has no proven starting point: the only t=0 solution, h=ℓ, is not in the space H defined by the orthogonality condition, so the set I is not shown to be nonempty. That is a load-bearing gap.\n\nWhat the paper does well: it defines the capillary Orlicz combination and capillary Orlicz surface area measure cleanly, and proves the capillary Orlicz-Minkowski and Orlicz-Brunn-Minkowski inequalities using the Alexandrov-Fenchel inequality and Jensen. I checked the algebra in those proofs and they hold up. The a priori estimates in Section 4 are routine but appear sound for the normalized equation (4.1).\n\nThe problems: first, H's orthogonality condition fails for h=ℓ. The weight in Definition 1.2 and in H becomes the constant a = φ'(1)/φ(1) − n − 1. If a ≠ 0, the premise is that v has zero mean, and v = ℓ − mean(ℓ) has zero mean but ∫ ℓ v = Var(ℓ) > 0. If a = 0, the premise holds for all v and taking v=ℓ gives ∫ ℓ² > 0. So ℓ ∉ H, and since no other t=0 solution is offered, the base case of the continuity method is missing. This is not a minor technicality; it is the existence proof's foundation.\n\nSecond, the estimates in Section 4 are for the volume-normalized equation (4.1), but the continuity equation (5.1) is unnormalized. The paper says this is easy to handle, but no argument is given, and it is not immediate because solutions along the path can have volume different from 1.\n\nThird, the remark that for φ(x)=x^p the orthogonality condition is trivial is wrong: for p=n+1 the coefficient vanishes and the condition would force ∫ h v = 0 for all v, which is impossible.\n\nWho this is for: convex geometers working on capillary problems and Orlicz-Brunn-Minkowski theory. The definitions and inequalities are worth having; the existence theorem as written is not.\n\nRecommendation: send it to peer review, but the referee should be asked to check the base case of the continuity method and the normalization transfer. The authors can likely repair the proof by reworking the solution space or the path, but the paper in its current form does not establish Theorem 1.1.","headline":"Natural capillary Orlicz theory with solid inequalities, but the main existence theorem has a missing starting point: the t=0 solution is not in the solution space H.","tokens_in":20171,"tokens_out":4909,"would_cite":true,"duration_ms":38612,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["53C42","53C45","35J66"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves existence of smooth symmetric volume-one solutions to the capillary Orlicz-Minkowski problem, with equality in the lower bound forcing a spherical cap.","keywords":["capillary hypersurface","capillary Orlicz-Minkowski problem","Robin boundary value condition","capillary Orlicz-Brunn-Minkowski inequality","Monge-Ampere equation","continuity method","spherical cap","convex geometry"],"falsifier":"Compute the orthogonality integral for $h=\\ell$ with $v=\\ell-\\bar{\\ell}$, where $\\bar{\\ell}$ is the average of $\\ell$ over $C_\\theta$. When $\\phi'(1)\\neq(n+1)\\phi(1)$, the premise in (1.3) holds because $\\int v=0$, yet $\\int \\ell v=\\int(\\ell-\\bar{\\ell})^2>0$, so the orthogonality condition fails; when $\\phi'(1)=(n+1)\\phi(1)$, the premise holds for every $v$, and taking $v=\\ell$ gives $\\int \\ell^2>0$. Thus $h=\\ell$ is not in $H$, and this single calculation decides whether the continuity method has a starting solution.","tokens_in":19053,"feed_emoji":"📐","tokens_out":13847,"duration_ms":100460,"temperature":0.7,"pith_summary":"The paper introduces a Robin-boundary analogue of the Orlicz-Minkowski problem: it asks for a convex body meeting the upper half-space at a fixed contact angle whose capillary Orlicz surface area measure is a prescribed positive even function on the spherical cap. The main theorem asserts that for every admissible weight function, satisfying mild growth and convexity conditions, and every even positive datum satisfying an integral lower bound, there is a smooth symmetric capillary convex body of volume one whose support function solves the associated Monge-Ampere equation with Robin boundary condition. If the lower bound is an equality and the weight is strictly convex, the only solution is the spherical cap. This matters because it extends the classical Minkowski and Orlicz-Minkowski problems to the capillary setting and, as a corollary, yields volume-normalized supercritical capillary Lp solutions with the normalization constant forced to 1.","feed_headline":"Existence proved for capillary Orlicz-Minkowski problem","feed_subtitle":"Prescribed capillary Orlicz surface area yields volume-one convex bodies; the equality case is a spherical cap.","key_machinery":"The load-bearing object is the capillary Orlicz surface area measure $dS^c_\\phi(b\\Sigma,\\xi)=\\phi(\\ell/h)\\,h\\,\\det(h_{ij}+h\\delta_{ij})\\,d\\xi$ on the spherical cap $C_\\theta$, which converts the geometric prescription into a Robin-boundary Monge-Ampere equation. The proof mechanism is the one-parameter continuity family with $f_t=(1-t)\\phi(1)\\ell+tf$; the solution space $H$ is defined by the orthogonality condition (1.3), which makes the linearized operator $L_h$ self-adjoint and surjective on $H$ through Lemmas 5.1 and 5.2. The a priori estimates of Section 4 close the method: the $C^0$ estimate uses the growth of $\\phi$ and the Steiner point argument for even functions, the $C^1$ estimate uses an auxiliary function with the distance-to-boundary term, the $C^2$ estimate uses maximum principles on $P=\\nabla^2_{\\Xi,\\Xi}h+h$ and a boundary auxiliary function $Q$, and bootstrap regularity then gives all higher norms. On the geometric side, the Orlicz-Minkowski inequality follows from Jensen's inequality together with the Aleksandrov-Fenchel inequality, and it supplies the equality case identifying the spherical cap.","core_discovery":"On the paper's own terms, the central discovery is that the prescribed-capillary-Orlicz-measure problem is solvable in the smooth category. Writing $h$ for the capillary support function on $C_\\theta=\\{\\xi\\in\\mathbb{R}^{n+1}_+: |\\xi-\\cos\\theta\\,e|=1\\}$, with $\\ell(\\xi)=\\sin^2\\theta+\\cos\\theta\\langle\\xi,e\\rangle$, the problem is the Robin-boundary Monge-Ampere equation $$\\$\\varphi$(\\ell/h)\\,h\\,\\det(h_{ij}+h\\delta_{ij})=f\\ \\text{in }C_\\$\\theta$,\\qquad \\nabla_\\mu h=\\cot\\$\\theta$\\,h\\ \\text{on }\\partial C_\\$\\theta$.$$ Theorem 1.1 asserts that for every $\\phi\\in\\mathcal{O}$ and every even positive $f\\in C^2(C_\\theta)$ satisfying the lower bound (1.4), this equation has a smooth, symmetric solution of volume one which also satisfies the orthogonality condition with respect to $f$; if equality holds in (1.4) and $\\phi$ is strictly convex, the solution is the spherical cap $|bC_\\theta|^{-1/(n+1)}C_\\theta$. The proof is a continuity method in the interpolating family $f_t=(1-t)\\phi(1)\\ell+tf$: a priori $C^0,C^1,C^2$ and higher estimates control solutions, the linearized operator is symmetric on the constrained space $H$, its kernel is characterized by Lemma 5.2, and the implicit function theorem gives openness, making the solvable set both open and closed. The companion capillary Orlicz-Minkowski and Orlicz-Brunn-Minkowski inequalities carry the equality and rigidity part.","pith_inferences":["The base step of the continuity method is not verified as written: at $h=\\ell$ and $t=0$, the premise in (1.3) holds for every $v$ with $\\int_{C_\\theta} v=0$ (or for every $v$ when $\\phi'(1)=(n+1)\\phi(1)$), while $\\int_{C_\\theta}\\ell v$ need not vanish, so $\\ell\\notin H$ and the parameter set $I$ may be empty at $t=0$.","Because the only role of $H$ is to make the linearized operator $L_h$ surjective, a transversality condition that $\\ell$ does satisfy could replace the orthogonality condition and likely restore the base step while preserving the openness argument.","The evenness assumption drives the $C^0$ estimate by forcing the Steiner point to the origin; non-even data would require a different normalization or a translation argument, so the theorem as stated does not cover them."],"forward_implications":["For $\\phi(x)=x^p$ with $p\\ge n+1$, Theorem 1.1 gives smooth symmetric volume-one solutions to the capillary even $L^p$-Minkowski problem; in the critical case $p=n+1$ the normalization constant $\\gamma$ from [46] is forced to be 1.","Equality in the lower bound (1.4) with strictly convex $\\phi$ pins the solution uniquely: the spherical cap $|bC_\\theta|^{-1/(n+1)}C_\\theta$ is the only volume-one capillary Orlicz solution.","The capillary Orlicz-Brunn-Minkowski inequality (3.4) makes capillary Orlicz addition a subadditive operation on volumes, with equality precisely for dilates when $\\phi$ is strictly convex.","The a priori estimates in Lemmas 4.1-4.5 control solutions in every $C^{m+1,\\alpha}$ norm by the data, so higher-regularity and compactness statements follow for the same class of Robin Monge-Ampere data."],"supporting_citations":[{"why":"Provides the capillary support-function formalism, the Robin boundary identities (2.1)-(2.2), and the base a priori estimates for the capillary Minkowski equation reused in Section 4.","marker":"[44]"},{"why":"Supplies the Aleksandrov-Fenchel inequality for convex hypersurfaces with capillary boundary, which underlies the Orlicz-Minkowski and Orlicz-Brunn-Minkowski inequalities and their equality cases.","marker":"[48]"},{"why":"Introduces the Orlicz addition and its variational formula (Lemma 8.4), used to define capillary Orlicz combinations and Orlicz mixed volumes.","marker":"[15]"},{"why":"Gives the oblique-boundary regularity theory for Monge-Ampere equations used to pass from C2 estimates to C^{2,alpha} and higher estimates.","marker":"[34]"},{"why":"Defines capillary Lp surface area measures and the supercritical capillary Lp problem; the corollary for p at least n+1 and the constant gamma=1 refer to it.","marker":"[46]"},{"why":"Provides the even Orlicz-Minkowski problem and its Monge-Ampere formulation on the sphere, the model generalized here to the Robin setting.","marker":"[21]"},{"why":"Used for the standard interior gradient bound in the C1 estimate.","marker":"[11]"},{"why":"Used for bootstrap regularity from C^{2,alpha} to higher-order estimates.","marker":"[17]"}],"fun_headline_variants":["Smooth existence for capillary Orlicz-Minkowski","Capillary Orlicz-Minkowski existence proved","Orlicz-Minkowski with capillary boundary: existence","Robin-boundary Monge-Ampere: capillary existence"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The argument's load-bearing premise is that the spherical cap solution $h=\\ell$ belongs to the constrained solution space $H$ used for the continuity method; that condition rules $h=\\ell$ out, so the base case at $t=0$ is not actually established.","fun_headline_variants_meta":{"raw":{"variants":["Smooth existence for capillary Orlicz-Minkowski","Capillary Orlicz-Minkowski existence proved","Orlicz-Minkowski with capillary boundary: existence","Robin-boundary Monge-Ampere: capillary existence"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000268,"raw_usage":{"total_tokens":1641,"prompt_tokens":989,"completion_tokens":652,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":605,"completion_tokens_details":{"reasoning_tokens":587}},"tokens_in":605,"tokens_out":652,"duration_ms":354132,"temperature":1.0,"reasoning_tokens":587,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T15:53:37.459500+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the orthogonality integral for $h=\\ell$ with $v=\\ell-\\bar{\\ell}$, where $\\bar{\\ell}$ is the average of $\\ell$ over $C_\\theta$. When $\\phi'(1)\\neq(n+1)\\phi(1)$, the premise in (1.3) holds because $\\int v=0$, yet $\\int \\ell v=\\int(\\ell-\\bar{\\ell})^2>0$, so the orthogonality condition fails; when $\\phi'(1)=(n+1)\\phi(1)$, the premise holds for every $v$, and taking $v=\\ell$ gives $\\int \\ell^2>0$. Thus $h=\\ell$ is not in $H$, and this single calculation decides whether the continuity method has a starting solution.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the capillary support-function formalism, the Robin boundary identities (2.1)-(2.2), and the base a priori estimates for the capillary Minkowski equation reused in Section 4."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the Aleksandrov-Fenchel inequality for convex hypersurfaces with capillary boundary, which underlies the Orlicz-Minkowski and Orlicz-Brunn-Minkowski inequalities and their equality cases."},{"cited_title":"Gardner, D","cited_arxiv_id":null,"evidence_quote":"Introduces the Orlicz addition and its variational formula (Lemma 8.4), used to define capillary Orlicz combinations and Orlicz mixed volumes."},{"cited_title":"Lions, N.S","cited_arxiv_id":null,"evidence_quote":"Gives the oblique-boundary regularity theory for Monge-Ampere equations used to pass from C2 estimates to C^{2,alpha} and higher estimates."},{"cited_title":"Haberl, E","cited_arxiv_id":null,"evidence_quote":"Provides the even Orlicz-Minkowski problem and its Monge-Ampere formulation on the sphere, the model generalized here to the Robin setting."},{"cited_title":"Chou, X.-J","cited_arxiv_id":null,"evidence_quote":"Used for the standard interior gradient bound in the C1 estimate."},{"cited_title":"Gilbarg, N.S","cited_arxiv_id":null,"evidence_quote":"Used for bootstrap regularity from C^{2,alpha} to higher-order estimates."}],"review_version":2}