A P\'olya--Szeg\H{o} Theorem for Tangential Polygons

Changfeng Gui; Qinfeng Li; Yeyao Hu

A P\'olya--Szeg\H{o} Theorem for Tangential Polygons

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Pith's one-line read For tangential polygons of fixed area, the regular N-gon uniquely maximizes torsional rigidity.

desk verdict Real new theorem: regular N-gon maximizes torsional rigidity among equal-area tangential N-gons, with a clean variational proof; the flagged analytic estimate is softer than the reader thinks. read the letter →

arxiv 2607.28768 v1 pith:HWTLNZVD submitted 2026-07-30 math.AP

Changfeng Gui , Yeyao Hu , Qinfeng Li This is my paper · ORCID

classification math.AP MSC 35J2549Q1052A40

keywords torsionalrigiditytangentialpolygonregularmixedboundaryproblemstrictconcavityspectralmeasuredeficitdecompositionfirstDirichleteigenvalue

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

This paper proves that among all convex N-gons whose sides touch a common circle (tangential polygons), the regular N-gon uniquely maximizes torsional rigidity for a fixed area. Since every triangle is tangential, this includes a new, independent proof of the classical triangle case: the equilateral triangle maximizes torsion among all fixed-area triangles. The argument is not a symmetrization but a relaxation: the polygon is cut into 2N right triangles by joining the incenter to vertices and tangency points, and the gluing conditions across the cuts are discarded to get an upper bound by a sum of mixed Dirichlet–Neumann cell rigidities. The main analytic step is a sharp strict-concavity property of a function built from the cell rigidity, proved by representing the cell problem through spectral measures and a pointwise kernel inequality. The method also yields an explicit deficit decomposition separating angular and perimeter deviations, and this leads to a new proof of the strict increase of torsion along equal-area regular polygons and to an eigenvalue comparison criterion.

What carries the argument

The central object is the function g(α)=h(tan α) − (1/8)tan α, where h is the mixed torsional rigidity of a right-triangular cell with one Dirichlet side and two Neumann sides, and α∈(0,π/2) is the angle between the Neumann sides. The proof's main work is establishing that g is strictly concave, with a quantitative strong-concavity bound; this is done by representing h through a spectral measure (or its Galerkin approximants) and using a pointwise inequality for a rational kernel whose slack term is uniformly positive on compact angle ranges. This concavity, combined with the mixed-cell relaxation bounding T(P) by a sum of cell rigidities, yields the extremal theorem via Jensen's inequality.

What would settle it

Numerically compute the torsional rigidity of fixed-area tangential quadrilaterals, for instance rhombi with varying vertex angles; any non-square rhombus whose torsion exceeds that of the equal-area square would refute Theorem 1.1. Equivalently, evaluate the second derivative of g(α)=h(tan α)−(1/8)tan α on a dense grid; a single point with nonnegative second derivative would contradict the concavity that the proof relies on.

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Extended reading notes

Core claim

The central claim is Theorem 1.1: if P is a convex tangential N-gon (N≥3) and R_N is the regular N-gon with the same area, then T(P) ≤ T(R_N), with equality if and only if P is regular. The proof uses a mixed-cell relaxation: the incenter to vertices and tangency points cuts P into 2N right triangles; dropping the matching conditions on the internal cuts gives T(P) ≤ 2r^4 ∑ h(a_i), where h is the torsional rigidity of a right-triangular cell with one Dirichlet side and two Neumann sides, and r is the inradius. The decisive estimate is the strict concavity of g(α)=h(tan α)−(1/8)tan α on (0,π/2), obtained by a Galerkin diagonalization that represents the cell problem as an integral against pos

Load-bearing premise

The proof's load-bearing premise is that a residual term in the central inequality, checked by explicit sign verification of a polynomial's coefficients, is uniformly positive on every compact angle range; if it could approach zero, the strict concavity and the uniqueness assertion would fail.

Editorial extensions

If this is right

The triangle case (N=3) gives an independent proof that the equilateral triangle uniquely maximizes torsional rigidity among all triangles of fixed area.
Torsional rigidity of equal-area regular polygons increases strictly with side count: T(R_N) < T(R_{N+1}) for all N ≥ 3, with an explicit positive lower bound on the gap.
The torsional deficit of a tangential polygon relative to its equal-area regular polygon is at least the sum of a nonnegative angular-asymmetry term and a nonnegative perimeter-excess term; the deficit controls the squared angular deviations when angles stay away from collapse.
A sufficient condition is derived: if the deficit term D_N(P) reaches a certain threshold (determined by the regular polygon's spectral slack), then the first Dirichlet eigenvalue of P exceeds that of the equal-area regular polygon.
As N grows, any potential counterexample to the eigenvalue extremal conjecture within the tangential class must lie within a relative perimeter band of width O(N^{-2}) around the regular polygon.

Reading between the lines

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

The mixed-cell relaxation could be applied to other energies (e.g., p-torsion or capacity), provided the corresponding cell functional satisfies a similar concavity; the method is not specific to torsion.
The spectral-measure representation behind the concavity proof may generalize to cone-like or higher-dimensional cells, potentially yielding isoperimetric results for revolution or cone-shaped domains.
The explicit deficit decomposition suggests a route to quantitative stability estimates for the full polygon problem: if a general N-gon can be approximated by a tangential one with controlled error, the two-term deficit may yield a measure of distance to regularity.
The paper's analytic proof of monotonicity along regular polygons might be adapted to show strict monotonicity of other shape functionals (such as the first eigenvalue) along regular polygons, if an analogous concavity for the eigenvalue cell function can be established.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Read the paper. The main theorem is real and new: for every N≥3, among convex tangential N-gons of fixed area, the regular N-gon uniquely maximizes torsional rigidity. Since every triangle is tangential, this gives a fresh proof of the classical triangular Pólya–Szegő result. The proof does not build on the known cases; it constructs a mixed-cell relaxation (cut the polygon at the incenter, discard transmission conditions) and then proves a sharp concavity property for a one-parameter mixed torsion problem. The Galerkin/spectral representation in Section 3 is sound, and the kernel inequality (3.32) is explicit enough to check. I spot-checked the algebra in (3.30)–(3.34) and it is consistent.

The novelty is substantial. The strict angular concavity theorem and the deficit decomposition are new, and the monotonicity along regular polygons (Corollary 1.3) is re-derived as a by-product. The paper is honest: it states clearly what is proven, what is a relaxation, and where it relies on prior work.

Soft spots, in proportion. The computation behind (3.35)—uniform positivity of the slack R on compact intervals—is lengthy and not machine-checked. But it is not load-bearing for the main theorem. For Theorem 1.1, only concavity of g is needed, which follows from pointwise R>0; the equality characterization comes from (4.5) forcing z=1 and then Jensen on the strictly convex tan. The uniform bound (3.35) is only needed for the quantitative strong concavity (3.1) and the stability estimate (5.3), which are interesting but secondary. So the reader's weakest-assumption flag is a real gap in verification but not a threat to the central result. Section 7 uses the authors' own prior expansion [9] and the Berghaus et al. eigenvalue expansion; that is peripheral to the main theorem but should be checked if someone wants to rely on the eigenvalue criterion.

Who this is for: anyone working on shape optimization or isoperimetric inequalities for torsion; it deserves a serious referee. I would send it out; the referee should verify the polynomial positivity but should not expect the main theorem to collapse.

Referee Report

0 major / 5 minor

Summary. The paper proves that, among all convex tangential N-gons of prescribed area, the regular N-gon uniquely maximizes torsional rigidity (Theorem 1.1). The proof decomposes a tangential polygon into 2N mixed Dirichlet–Neumann right-triangular cells, relaxes the transmission conditions, and reduces the problem to a sharp strict-concavity property of g(α)=h(tan α)−(1/8)tan α. This concavity is established through a Galerkin/spectral representation of a one-dimensional minimization and an explicit rational-kernel inequality. As consequences, the paper obtains the triangular Pólya–Szegő theorem (Corollary 1.2), a quantitative deficit decomposition, a new proof of strict monotonicity of torsional rigidity along equal-area regular polygons (Corollary 1.3), and a sufficient condition, via the Kohler–Jobin inequality, for a tangential polygon to have larger first Dirichlet eigenvalue than the equal-area regular polygon.

Significance. If the result stands, it settles the polygonal Pólya–Szegő conjecture for the entire tangential class for every N≥3, a natural and nontrivial restricted class that includes all triangles. The proof is essentially self-contained for the main theorem: the variational relaxation, the projection lemma, the finite-dimensional Galerkin diagonalization, and the kernel inequality are all explicit and checkable. The equality characterization does not rely on the quantitative strong-concavity bound and follows from the perimeter term and strict convexity of tan. The paper also gives a constructive deficit decomposition separating angular asymmetry from perimeter excess, which is a useful quantitative tool. Corollary 1.3 provides an independent route to the recent monotonicity result. The applications in Section 7 are cleanly separated from the core proof and use cited asymptotic expansions appropriately. A particular strength is that the main analytic step is reduced to an explicit rational inequality with positive coefficients, which can be verified by direct algebra.

minor comments (5)

[Section 3, spectral-measure formulation] After (3.38), the measure μ is defined for Borel subsets B⊂(0,1), but the subsequent integral is taken over (0,∞). This is surely a typo and should be corrected to B⊂(0,∞).
[Section 3, proof of (3.35)] The verification of the uniform positivity of R(x,λ) is terse. It would help to state explicitly that one fixes ε>0, uses the uniform limits as λ→0 and λ→∞ on [x0,x1] to bound the tails, and then applies compactness on [x0,x1]×[ε,1/ε]. The argument is correct as written, but the rectangle to which compactness is applied is left implicit.
[Section 6] The continuity statement (6.1) for torsional rigidity under Hausdorff convergence of convex domains is used without a reference. Adding a citation or a one-line justification would improve self-containedness.
[Section 7, Lemma 7.2] The proof relies on the expansions (7.8) and (7.9) from [9] and [2]. Since the cancellation of the N^{-4} term in the eigenvalue expansion is essential for the claimed leading-order behavior, it would be useful to state explicitly that [2, Eq. (2)] indeed has no N^{-4} term, or to quote the expansion to the required order.
[Throughout] There are a few minor typographical issues in the typeset version (e.g., broken words in the title/abstract, the overline in bH in the spectral section). These do not affect the mathematics and can be fixed during production.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: Theorem 1.1 is derived from scratch; the only self-citation is a peripheral asymptotic input in Sec. 7.

full rationale

I walked the derivation chain. Theorem 1.1 is proved from Proposition 2.1 (mixed-cell relaxation) and Theorem 3.1 (strict concavity of g), both of which are established in-paper from the definition of h(a) via completion of the square, zero-mode removal, Galerkin spectral representation, the moment bounds (3.27), and the kernel inequality (3.32)-(3.36). No step invokes the classical triangular Pólya–Szegő theorem or the polygonal conjecture; the equality case uses only (4.5), the strict convexity of tan, and the equality case of the relaxation, not the theorem being proved. Corollary 1.3 is proved independently by applying the deficit estimate (Proposition 5.1) to a degenerating tangential (N+1)-gon; it does not use [9] in its proof. The sole self-citation is [9, Theorem 1.4] in Lemma 7.2, which supplies the regular-polygon torsion expansion used for the asymptotic Kohler–Jobin slack. This is not load-bearing for the main extremal theorem, and it is not the target result being derived, so it does not create a circular chain. The alleged weakest point (3.35) is stronger than needed for the inequality; only nonnegativity of R is needed for concavity, and the uniqueness follows from the equality analysis, not from strong concavity.

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

No free parameters or invented entities. The only external inputs are standard functional analysis and two cited asymptotic results (appearing only in Section 7). The main theorem is derived from first principles via the mixed-cell relaxation and the self-contained concavity proof.

assumptions (5)

standard math Kohler–Jobin inequality: λ1(Ω)² T(Ω) ≥ π j_{0,1}⁴/8, strict unless Ω is a disk.
Used in Section 7 (Proposition 7.1) to convert a torsional deficit into an eigenvalue comparison. Cited from [5].
domain assumption Regular-polygon asymptotic expansions from [9, Theorem 1.4] and [2, Eq. (2)] for T(R_N) and λ1(R_N).
Used only in Lemma 7.2 to estimate η_N(A). These are external results (one from the authors' own prior work [9]); their correctness is not load-bearing for the main theorem but is needed for the asymptotic exclusion statement in Section 7.
standard math Standard Sobolev/trace theory, Riesz representation, Kato's second representation theorem, and the spectral theorem.
Used throughout Section 3 for the variational formulation, the projection lemma, and the spectral-measure formulation; standard functional analysis.
standard math Continuity of torsional rigidity under Hausdorff convergence of convex domains.
Used in the proof of Corollary 1.3 to pass to the limit ε→0.
standard math Strict convexity of tan on (0,π/2).
Used in (4.2) to get S ≥ N tan(π/N) with equality only for equal angles.

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Pith. "Pith review of A P\'olya--Szeg\H{o} Theorem for Tangential Polygons." pith.science (2026). https://pith.science/paper/HWTLNZVD

@misc{pith2026260728768,
  author       = {Pith},
  title        = {Pith review of: A P\'olya--Szeg\Ho Theorem for Tangential Polygons},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/HWTLNZVD}},
  note         = {Machine review of arXiv:2607.28768}
}

abstract

We prove that, for every integer $N\ge3$, the regular $N$-gon uniquely maximizes torsional rigidity among all tangential $N$-gons of prescribed area. Since every triangle is tangential, the case $N=3$ yields an independent proof of the classical triangular P\'olya--Szeg\H{o} theorem. The proof decomposes a tangential polygon into mixed Dirichlet--Neumann right-triangular cells. Its analytic core is the strict concavity of \[ \alpha\longmapsto h(\tan\alpha)-\frac18\tan\alpha, \] where $h$ is the mixed torsional rigidity of a right-triangular cell with one Dirichlet side and two Neumann sides, and $\alpha\in(0,\pi/2)$ is the angle between the Neumann sides. We also obtain an explicit deficit decomposition that separates angular asymmetry from perimeter excess. As applications, we give a novel analytic proof that the torsional rigidity of equal-area regular polygons increases strictly with the number of sides, and derive an explicit criterion ensuring that a tangential polygon has larger first Dirichlet eigenvalue than the equal-area regular polygon.

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Reference graph

Works this paper leans on

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Pith tools

Reviewed August 3, 2026 · model on record in the stance chip above.

[1] [9]

C. Gui, Y. Hu, Q. Li, and C. Zhang, Mixed torsion on right triangles and the Pólya–Szegő monotonicity problem for regular polygons , arXiv preprint arXiv:2606.13448, 2026

arXiv 2026

[2] [1]

Antunes and P

P. Antunes and P. Freitas, New bounds for the principal Dirichlet eigenvalue of planar regions , Experiment. Math. 15 (2006), no. 3, 333–342

2006

[3] [2]

Berghaus, B

D. Berghaus, B. Georgiev, H. Monien, and D. Radchenko, On Dirichlet eigenvalues of regular polygons , J. Math. Anal. Appl. 538 (2024), no. 2, Paper No. 128460, 17 pp

2024

[4] [3]

Bogosel and D

B. Bogosel and D. Bucur, On the polygonal Faber–Krahn inequality, J. Éc. polytech. Math. 11 (2024), 19–105

2024

[5] [4]

Bogosel and D

B. Bogosel and D. Bucur, Polygonal Faber–Krahn inequality: local minimality via validated computing , arXiv preprint arXiv:2406.11575, 2024

arXiv 2024

[6] [5]

Brasco, On torsional rigidity and principal frequencies: an invitation to the Kohler–Jobin rearrangement technique, ESAIM Control Optim

L. Brasco, On torsional rigidity and principal frequencies: an invitation to the Kohler–Jobin rearrangement technique, ESAIM Control Optim. Calc. Var. 20 (2014), no. 2, 315–338

2014

[7] [6]

Bucur and I

D. Bucur and I. Fragalà, Symmetry results for variational energies on convex polygons , ESAIM Control Optim. Calc. Var. 27 (2021), Paper No. 3, 17 pp

2021

[8] [7]

Dahne, J

J. Dahne, J. Gómez-Serrano, and J. Pech-Alberich, Monotonicity of the first Dirichlet eigenvalue of regular polygons, arXiv preprint arXiv:2601.16285, 2026

arXiv 2026