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REVIEW 3 major objections 5 minor 41 references

On the stability of the annulus for the torsion of multiply connected domains

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Pith's one-line read A plane domain whose torsional rigidity is nearly that of the optimal annulus must itself be nearly an annulus, with a quantitative power-law bound on the symmetric-difference distance.

desk verdict The main gap flagged by the reader dissolves on a second pass; this is a solid quantitative stability result for the Pólya–Weinstein annulus inequality, with typographical presentation problems that a good revision should clean up. read the letter →

arxiv 2506.07592 v2 pith:TYP7O2NW submitted 2025-06-09 math.AP math.SP

Vincenzo Amato , Luca Barbato This is my paper
classification math.APmath.SP MSC 35J0535B3535B45
keywords torsionalrigiditymultiplyconnecteddomainsannulusquantitativeisoperimetricinequalitystabilitysymmetrizationPólya–WeinsteinFraenkelasymmetry
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 proves a quantitative stability theorem for the classical isoperimetric problem of beam torsion: among plane domains with a fixed area and a fixed total area of holes, the concentric annulus maximizes torsional rigidity, and the paper establishes a converse in quantitative form. If a multiply connected domain's torsional rigidity is within $\varepsilon$ of the annular optimum, then the domain must lie within a power of $\varepsilon$ of some annulus, measured by symmetric difference of sets. The proof works by showing that the torsion function of the domain is close, in $L^1$, to its symmetric rearrangement, so that its superlevel sets are close to concentric balls; separate arguments control the outer and inner boundaries. A corollary is rigidity: equality in the Pólya–Weinstein inequality is attained only by an annulus, up to translation. This gives a direct link between a physical quantity, torsional stiffness, and a geometric one, annular shape.

What carries the argument

The engine is the comparison-by-symmetrization machinery: the torsion function $u$ of $\Omega$ is extended by constants on the holes, and a rearrangement comparison gives $\tilde u^{\sharp}\le \widetilde V$ pointwise, where $V$ is the explicit torsion function of the optimal annulus, with $|\nabla \widetilde V(x)|=|x|$. A truncated function $w=\min\{\tilde u,\tilde u^{*}(|S|)\}$ is introduced; its Pólya–Szegő deficit is controlled by the torsion deficit $T(\Omega_{\mathrm O})-T(\Omega)$, and the quantitative Pólya–Szegő inequality then forces $w$ to be close to its rearrangement $w^{\sharp}$ in $L^1$. The set where $\nabla w^{\sharp}$ is small is decomposed into level ranges: outside a small exceptional set of levels the symmetrization arguments control the measure directly, and the exceptional levels are shown to be few by an isoperimetric estimate combined with the Fleming–Rishel formula. The almost-radiality of $w$ is transferred to $\tilde u$, and then to its superlevel sets, yielding outer and inner balls that are close and concentric.

What would settle it

Compute, for a family of near-annular domains such as an annulus with a small off-centre hole, the level sets $\{u>t\}$ and check whether any of them, for $t$ outside the exceptional interval $I$ of Lemma 6.5, lies entirely inside a hole; if such levels form a set of positive measure, the estimate (47) in the proof fails. A direct contradiction of Theorem 1.3 would be a sequence $\Omega_n$ with fixed $|G|$ and $|S|$ such that $T(\Omega_{\mathrm O})-T(\Omega_n)\to 0$ while $\beta(\Omega_n)\ge c>0$.

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Extended reading notes

Core claim

The central claim is Theorem 1.3: for every multiply connected domain $\Omega = G\setminus S$ in the plane, there are constants $C>0$ and $\theta>0$ depending only on $|G|$ and $|S|$ such that $$T(\Omega_{\mathrm O})-T(\$\Omega$)\ge C\,\$\beta$(\$\Omega$)^{\$\theta$},$$ where $T$ is torsional rigidity, $\Omega_{\mathrm O}$ is the concentric annulus with $|G^{\sharp}|=|G|$ and $|S^{\sharp}|=|S|$, and $\beta(\Omega)=\inf\{|\Omega\triangle A|: A=B_1\setminus B_2,\ |B_1|=|G|,\ |B_2|=|S|\}$ is the annular asymmetry index. The proof also yields an explicit estimate, when $|S|\le \tfrac23|G|$, of the form $$T(\Omega_{\mathrm O})-T(\$\Omega$)\ge \frac{c}{\pi\$gamma_2^{2}$}\bigl(|G|^2\$\alpha$(G)^3+|S|^2\$\alpha$(S)^3\bigr),$$ where $\alpha$ is the Fraenkel asymmetry index and $\gamma_2$ the two-dimensional quantitative isoperimetric constant. In particular, a small deficit forces the outer boundary to be close to a circle and the holes to be close to a ball of the right area, and the two balls to be nearly concentric. Equality in the underlying Pólya–Weinstein inequality is characterized: it holds only for the annulus itself, up to translation.

Load-bearing premise

The proof of Lemma 6.5 relies on the assertion that, for all level values outside a small exceptional set, the superlevel sets of the torsion function cannot lie entirely inside a hole because the gradient of $u$ is close to that of $V$; this closeness is assumed rather than derived from the small torsion deficit.

Editorial extensions

If this is right

  • If Theorem 1.3 is correct, then the torsion gap provides a genuine geometric distance: a beam whose rigidity is within $\varepsilon$ of the annular maximum must have a cross-section within $C\varepsilon^{\theta/2}$, in symmetric difference, of some annulus with the same total area and hole area.
  • The equality-case characterization closes the rigidity question for the Pólya–Weinstein inequality: among multiply connected plane domains with fixed $|G|$ and $|S|$, the annulus is the unique maximizer up to translation.
  • Theorem 1.2 gives the first explicit algebraic rate, with exponent $3$ in the Fraenkel asymmetries of the outer domain and the holes, under the mild restriction $|S|\le \tfrac23|G|$.
  • The proof shows that the torsion function itself is almost radial: $\tilde u$ is close in $L^1$ to $\tilde u^{\sharp}$, and this near-radiality is the mechanism behind the geometric conclusion.

Reading between the lines

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

  • The rigidity part already works in any dimension and for degenerate elliptic operators, so the same quantitative route may yield a stability estimate for the generalized torsion functional $T(\Omega,a_{ij})$ with constants depending on the weight function $\nu$.
  • The restriction $|S|\le \tfrac23|G|$ in Theorem 1.2 is probably an artifact of the threshold argument: as $|S|$ approaches $|G|$ the torsion itself becomes tiny, so a version with constants depending on $|G|-|S|$ might cover the full range.
  • A testable extension is to compute the optimal exponent $\theta$ for the explicit family of eccentric annuli, two circles with fixed areas and varying center distance; for small eccentricity $\delta$, both the torsion deficit and $\beta$ should scale as powers of $\delta$, giving a numerical value for the best possible $\theta$.
  • If the gradient-closeness assertion inside Lemma 6.5 fails for some near-annular configuration, the almost-radiality theorem could still be recovered by a different bound on the set of level values whose superlevel sets sit inside holes; locating such a configuration would pinpoint exactly which part of the symmetrization argument needs repair.
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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 / 5 minor

Summary. The paper establishes a quantitative stability theorem for the Pólya-Weinstein isoperimetric inequality for the torsional rigidity of multiply connected planar domains. It proves that if the torsional rigidity of a multiply connected domain Ω is close to the optimal value attained by the annulus with the same outer area and same total hole area, then Ω must be close, in a symmetric-difference sense, to an annulus (Theorem 1.3). The proof combines a quantitative isoperimetric inequality, a quantitative Pólya-Szegő principle, and a propagation-of-asymmetry technique. It also proves a rigidity result (Proposition 1.1) for the equality case, and an explicit quantitative bound in terms of the Fraenkel asymmetries of the outer boundary and of the holes (Theorem 1.2).

Significance. If the result is correct, the paper is a substantial contribution: the equality case for the Pólya-Weinstein inequality was previously characterized only partially, and no quantitative stability statement for the annulus in terms of an annular asymmetry index appears to have been known. The proof strategy is well chosen and mostly follows established techniques, and the paper gives explicit, if non-sharp, constants in Theorem 1.2. The dependence on non-explicit constants from the quantitative Pólya-Szegő principle is acknowledged and is not circular. The two main technical gaps identified below are local and appear repairable without changing the overall strategy.

major comments (3)
  1. [Lemma 6.2] The proof of Lemma 6.2 rests on the asserted bound ∫_{G^#}|∇\tilde V|^2 - ∫_{G^#}|∇w^#|^2 ≤ 6|G|√(2π) ε^{1/2}. This does not follow directly from Lemma 6.1, which controls E_w - E_{w^#}, not E_V - E_{w^#}. The missing ingredient is a bound on E_top = ∫_{\{u>u^*(|S|)\}}|∇u|^2. Such a bound can be obtained from the layer identity ∫_{\{u=t\}}|∇u| dH = μ_{\tilde u}(t), the pointwise comparison μ_{\tilde u}(t) ≤ μ_{\tilde V}(t), and the unconditional inequality 2π(\tilde V(|S|)-u^*(|S|))^2 ≤ T(Ω_O)-T(Ω) obtained from the test function f in (37). Since the estimate on |I| is used in Lemma 6.5 and Proposition 6.6, this missing derivation is load-bearing and should be supplied.
  2. [Lemma 6.4] After equation (46), the proof uses the inequality |x| ≤ |∇\tilde V|(ℓ(t)x). By definition (10) in dimension two, |∇\tilde V|(y) = |y|/2 = ℓ(t)|x|/2, so this inequality requires ℓ(t) ≥ 2. Only ℓ(t) ≥ 1 follows from μ_{\tilde V}(t) ≥ μ_{\tilde u}(t), and ℓ is close to 1 in the near-optimal regime. The argument can be repaired by replacing the displayed inequality with the weaker |x|/2 ≤ |∇\tilde V|(ℓ(t)x), which still gives |x| ≤ 2δ + O(ε^{1/8}) and proves the lemma with slightly enlarged constants. The proof as printed should be corrected.
  3. [Theorem 1.2 and Section 5] The constants in Theorem 1.2 and Proposition 5.5 do not match. Proposition 5.5 gives a bound with denominator containing 3^2·2^{10}, while Theorem 1.2 states a denominator with 3^2·2^9. Since Proposition 4.2 has an even smaller denominator (2^8), the combined constant in Theorem 1.2 cannot be numerically larger than the minimum of the two constants. This is only a bookkeeping issue because the constants are not claimed to be sharp, but the theorem statement should be adjusted to the actual value that the proof yields.
minor comments (5)
  1. [Lemma 6.5] The sentence about the gradient of u being close to that of V is unnecessary and confusing. The inequality -μ'_u(t) ≥ -μ'_V(t) follows from the displayed isoperimetric chain and the fact that ζ^{-1} ≤ 1; the proof holds for all relevant levels without invoking any unproved gradient-closeness assertion.
  2. [Equation (45)] Equation (45) appears to typeset a quotient where a product is intended; it should read P(\tilde V=t)|∇\tilde V|(y) - P(w^#=t)|∇w^#|(x) ≤ ε^{1/4}.
  3. [Lemma 4.1] In the measure estimate, the expression ũ^*(s_G) is dimensionally inconsistent because s_G is a level height rather than a rearrangement index. The intended quantity appears to be the level s_G itself, so the notation should be clarified.
  4. [Section 5 constants] Several constants in Section 5 are printed in compressed form such as '3224πγ2' and '32210πγ2', which makes verification unnecessarily difficult. Please write them as 3^2 2^4 π γ^2 and 3^2 2^{10} π γ^2, respectively.
  5. [Assumptions on |S|/|G|] The paper uses both |S| ≤ 3/4 |G| (in Proposition 4.2) and |S| ≤ 2/3 |G| (in Theorem 1.2 and Lemma 5.2). Since 2/3 < 3/4, this is harmless, but the relationship should be stated explicitly to avoid the appearance of a mismatch.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the stability bound is derived from external quantitative isoperimetric and rearrangement inequalities; the target inequality is not used as an input.

full rationale

The paper's derivation chain is self-contained against external benchmarks. The main results, Theorems 1.2 and 1.3, are proved by combining (i) the Pólya–Weinstein symmetrization bound for multiply connected domains (Buonocore, external), (ii) the sharp quantitative isoperimetric inequality (Figalli–Maggi–Pratelli, external), (iii) the quantitative Pólya–Szegő principle (Cianchi–Esposito–Fusco–Trombetti, external), and (iv) a level-set asymmetry propagation argument attributed to Hansen–Nadirashvili and Brasco–De Philippis, not to the present authors. No parameter appearing in the final inequality is fitted to the quantity being predicted: the deficit T(Ω_O) − T(Ω) and the annular asymmetry index β(Ω) are defined independently, and the proof concludes by deriving β(Ω) ≤ C ε^{θ/2} from ε = T(Ω_O) − T(Ω), which is the contrapositive form of Theorem 1.3. The self-citations [3, 4, 5] are cited only as examples of prior applications of the propagation technique, and they are not load-bearing: the actual estimates used are quoted from the external references [14, 23, 30]. The assertion in Lemma 6.5 that superlevel sets cannot lie entirely inside a hole because the gradient of u is close to that of V is not an input to the theorem and is not a fitted assumption; it is, at worst, a potential mathematical gap, not a circular step. No equation in the paper reduces to another by construction, and no fitted input is renamed as a prediction. The result therefore exhibits no significant circularity.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The central claim rests entirely on established quantitative isoperimetric and rearrangement theorems, plus explicit size assumptions. No free parameters are fitted and no new entities are introduced.

assumptions (5)
  • standard math Quantitative isoperimetric inequality (Theorem 2.9)
    Used to measure asymmetry of superlevel sets of the torsion function; constant γ_n from Figalli-Maggi-Pratelli [23].
  • standard math Quantitative Pólya-Szegő inequality (Theorem 2.11)
    Gives L1 closeness between a Sobolev function and its rearrangement; provides the non-explicit constants r,s that determine the exponent in Theorem 1.3.
  • standard math Brothers-Ziemer theorem (Theorem 3.2)
    Classifies equality in Pólya-Szegő; used in the rigidity proof to deduce that the outer domain is a ball.
  • standard math Buonocore/Pólya-Weinstein comparison (Theorem 2.6 and Corollary 2.8)
    The isoperimetric inequality being quantified; also provides pointwise comparison of rearranged torsion functions.
  • domain assumption Hole-size restriction |S| ≤ 2/3 |G| in Theorem 1.2
    Keeps the threshold t1 within the linear part of the annulus distribution function; the paper argues some such restriction is necessary for its proof strategy.

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Pith. "Pith review of On the stability of the annulus for the torsion of multiply connected domains." pith.science (2026). https://pith.science/paper/TYP7O2NW

@misc{pith2026250607592,
  author       = {Pith},
  title        = {Pith review of: On the stability of the annulus for the torsion of multiply connected domains},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/TYP7O2NW}},
  note         = {Machine review of arXiv:2506.07592}
}
abstract

We establish a quantitative version of the isoperimetric inequality for the torsion of multiply connected domains, among sets with given area and with given joint area of the holes. Since the optimal shape is the annulus, we investigate how a given domain approaches an annular configuration when its torsion is close to the optimal value. Our result shows that when the torsional rigidity is nearly optimal, the domain $\Omega$ must be close to an annulus.

Figures

Figures reproduced from arXiv: 2506.07592 by the authors.

Figure 1
Figure 1. Ω = B1 \ S. • |Br| < s < |BR| u(s) = n −1ω − 2 n n 2  |G| 2 n − |BR| 2 n  ; • s < |Br| u(s) = n −1ω − 2 n n 2  |G| 2 n − |BR| 2 n  + ˆ |Br| s t −1+ 2 n dt = n −1ω − 2 n n 2  |G| 2 n − |BR| 2 n + |Br| 2 n − s 2 n  . On the other hand, the solution to the symmetrized problem is: V (s) =    n −1ω − 2 n n 2 (|G| 2 n − s 2 n ) if s ≥ |S| n −1ω − 2 n n 2 (|G| 2 n − |S| 2 n ) if s < |S|. Therefore, by notici… view at source ↗
Figure 2
Figure 2. Upper bound for u˜ ∗ (s) Proof. Let B be the ball which realizes α(S˜). It is sufficient to observe that |S|α(S) ≤ |S△B| ≤ [PITH_FULL_IMAGE:figures/full_fig_p021_2.png] view at source ↗

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