Neumann analogue of the Pólya–Szegő polygonal eigenvalue conjecture

For every integer $n\ge 3$ and every convex planar $n$ -gon $P$ of area $A$ , let $\lambda_1^N(P)$ denote the first nonzero eigenvalue of the Laplace operator with Neumann boundary conditions on $P$ . If $R_n$ is the regular $n$ -gon of area $A$ , then $\lambda_1^N(P)\le \lambda_1^N(R_n)$ , with equality only when $P$ is congruent to $R_n$ .

References

Primary source

arXiv

Additional references

Maximizing the fundamental Laplace--Neumann eigenvalue on quadrilaterals — arXiv

geometric-analysis partial-differential-equations spectral-geometry

Posted 1 month ago

Progress summary

Refreshed 1 month ago

Claimed progress

A new paper claims the conjecture for four-sided shapes, while the cases with five or more sides remain open and the claim has not been independently verified.

The conjecture says that, among convex polygons with a fixed number of sides and area, the regular polygon uniquely gives the largest first nonzero Neumann eigenvalue. The triangle case is known, and the newly reported quadrilateral result addresses only $n=4$ .

Known results and September 2026 quadrilateral development

$n=3$ : the equilateral triangle is the unique maximizer among triangles.
September 2, 2026: Endo and Osting’s manuscript Maximizing the fundamental Laplace--Neumann eigenvalue on quadrilaterals claims that the square uniquely maximizes $\lambda_1^N$ among convex quadrilaterals of equal area.
The claimed proof uses Rayleigh--Ritz bounds, symmetry-reduced local analysis, and certified computer-assisted global estimates; it remains unverified.
No result settling the stated conjecture for $n\ge 5$ was found.

Community submission (unverified) — September 4, 2026

A submitted proof argues for a general framework using Rayleigh quotients, finite-dimensional approximation, dihedral symmetry, local Hessian analysis, and certified inequalities, but explicitly identifies analytical obstacles and does not claim a proof for arbitrary $n$ .

Current status (as of September 2026): The triangle case is known and the quadrilateral case is claimed solved by the square, but that claim is unverified and the conjecture remains open for $n\ge 5$ .

Sources

ProofLet P⊂R^2be a convex n-gon of area A, and let λ_1^N (P)denote the first nonzero eigenvalue of the Laplacian with Neumann boundary conditions. We investigate the conjecture that the regular n-gon uniquely maximizes λ_1^Namong convex n-gons of prescribed area.See full solutionHide full solution

Let P⊂R^2be a convex n-gon of area A, and let λ_1^N (P)denote the first nonzero eigenvalue of the Laplacian with Neumann boundary conditions. We investigate the conjecture that the regular n-gon uniquely maximizes λ_1^Namong convex n-gons of prescribed area. The conjecture is known for triangles, where the equilateral triangle is the unique maximizer, and for convex quadrilaterals, where recent work of Endo and Osting proves that the square is the unique maximizer. For n≥5, the conjecture remains open. We develop a systematic framework based on the Rayleigh quotient, similarity normalization, finite-dimensional Rayleigh–Ritz approximation, dihedral symmetry, local Hessian stability, exclusion of degenerating maximizing sequences, and certified global inequalities. We formulate a master finite-dimensional proposition whose verification would imply the full conjecture. We also identify the principal analytical obstacles that prevent the present framework from being regarded as a proof for arbitrary n.

Neumann_Polya_Szego_Polygonal_Eigenvalue_Manuscript.pdf379,462 bytesOpen
Solution_1.pdf539,612 bytesOpen
A Research Manuscript Toward the Neumann Polygonal Pólya_1article.pdf580,469 bytesOpen
The Neumann Analogue of the Pólya_manuscipt.pdf619,243 bytesOpen
Solution_1_report.pdf537,226 bytesOpen