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REVIEW 3 major objections 4 minor 59 references

The weighted isoperimetric inequality and Sobolev inequality outside convex sets

T0 review · 3 major / 4 minor · reviewed 2026-08-04 · deepseek-v4-flash

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Pith's one-line read 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

desk verdict 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. read the letter →

arxiv 2510.01647 v4 pith:2XB7PQRB submitted 2025-10-02 math.AP

Lu Chen , Jiali Lan This is my paper · ORCID
classification math.AP MSC 35A2335J9249Q2052A40
keywords capillaryisoperimetricinequalityweightedSobolevABPmethodhomogeneousweightsconvexsetsSchwarzsymmetrizationPólya-Szegőprinciple
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

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.

What carries the argument

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.

What would settle it

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.

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Extended reading notes

Core claim

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

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • 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.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 4 minor

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.

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 (3)
  1. [§1, Eq. (1.5); §3, Lemma 3.5, Eq. (3.27)] 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.
  2. [§1, Theorem 1.1; Definition 3.2/(3.23)] 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.
  3. [§4, proof of Theorem 1.1] 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.
minor comments (4)
  1. [§3, Lemma 3.5] 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.
  2. [§1, Theorems 1.3 and 1.4] 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.
  3. [§4, Definition 4.2] 'Kuratoswki' should be 'Kuratowski'.
  4. [Throughout] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the derivation is conditional on an explicit hypothesis and relies on external benchmarks; the vacuity of condition (1.5) is a correctness issue, not a circular one.

full rationale

The paper's central inequality is proved under an explicit λ_w-ABP hypothesis (Definition 3.2, eq. (3.23)) and a weight class (1.5). The proof uses the ABP method with a Neumann problem (1.9), Lemma 3.3, and the λ_w-ABP property as a black box; no parameter is fitted and no 'prediction' is renamed input. The half-space Sobolev constant is imported from the independent external paper [19] (Lemma 5.3), and the sharpness of the constant is checked against that external benchmark, so the Sobolev result is not circular. The condition w^{1/α} concave in (1.5) appears to be empty for positive α-homogeneous weights on all of R^n (taking y=-x in (3.27) gives α(w(-x)/w(x))^{1/α} ≤ -α), making Theorems 1.1, 1.2, 1.4 vacuous as stated; this is a serious correctness risk, but it is not a circularity of the derivation chain. Similarly, assuming the λ_w-ABP property for arbitrary convex E is a strong, unverified hypothesis (Remark 1.1 concedes the general convex case is open), but the paper explicitly labels it as an assumption rather than deriving it from the conclusion. No self-citation is load-bearing. Therefore the circularity score is 0.

Assumptions & free parameters 0 free parameters · 4 assumptions · 1 invented entities

The central load-bearing assumptions are the impossible weight condition (1.5) and the unverified lambda_w-ABP property. The external Sobolev and approximation results are legitimate inputs. No fitted parameters or new physical entities are introduced.

assumptions (4)
  • domain assumption The weight class in (1.5) is nonempty: w is positive, even, alpha-homogeneous, and w^{1/alpha} is concave on all of R^n.
    This is the main hypothesis of Theorems 1.1-1.4. As written, the class is empty: concavity of a positive one-homogeneous function forces v(-x) <= -v(x), impossible for v > 0. The paper's own Lemma 3.5 with y = -x gives alpha (w(-x)/w(x))^{1/alpha} <= -alpha.
  • ad hoc to paper The convex set E satisfies the lambda_w-ABP property (3.23).
    Introduced in Section 3 and used as the key estimate (3.19)/(3.23) in the proof of Theorem 1.1. The paper proves it only for half-spaces and gives no nontrivial convex example; Remark 1.1 says the general convex case remains open.
  • domain assumption The half-space weighted Sobolev inequality of Ciraolo-Figalli-Roncoroni (Lemma 5.3).
    Borrowed from [19]; it determines the sharp constant and extremals in Theorems 1.3-1.4. This is legitimate input but is an external theorem.
  • domain assumption Approximation lemmas from [2,31] (extension domains, Lemma 4.3, Lemma 4.4).
    Used in Section 4 to pass from smooth sets to sets of finite perimeter; proofs rely on the cited literature.
invented entities (1)
  • lambda_w-ABP property (3.23)
    purpose: Assumed geometric condition on the convex set E used to close the ABP estimate (3.33); it is the key unverified input for Theorem 1.1.
    The paper shows it holds automatically for half-spaces but provides no example for any other convex set; Remark 1.1 says the general case remains open.

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Pith. "Pith review of The weighted isoperimetric inequality and Sobolev inequality outside convex sets." pith.science (2026). https://pith.science/paper/2XB7PQRB

@misc{pith2026251001647,
  author       = {Pith},
  title        = {Pith review of: The weighted isoperimetric inequality and Sobolev inequality outside convex sets},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2XB7PQRB}},
  note         = {Machine review of arXiv:2510.01647}
}
abstract

In this paper, we establish a weighted capillary isoperimetric inequality outside convex sets using the $\lambda_w$-ABP method. The weight function $w$ is assumed to be positive, even, and homogeneous of degree $\alpha$, such that $w^{1/\alpha}$ is concave on $\R^n$. Based on the weighted isoperimetric inequality, we develop a technique of capillary Schwarz symmetrization outside convex sets, and establish a weighted P\'{o}lya-Szeg\"{o} principle and a sharp weighted capillary Sobolev inequality outside convex domain. Our result can be seen as an extension of the weighted Sobolev inequality in the half-space established by Ciraolo-Figalli-Roncoroni in \cite{CFR}.

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    T. Young,An essay on the cohesion of fluids, Philos. Trans. Roy. Soc. London95(1805), 65–87. 1 30 LU CHEN AND JIALI LAN (Lu Chen)Key Laboratory of Algebraic Lie Theory and Analysis of Ministry of Education, School of Mathematics and Statistics, Beijing Institute of Technology,...

Pith tools

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