Neumann polygon maximization conjecture

Establish whether the regular n-gon maximizes the first nonzero Laplace–Neumann eigenvalue among planar n-gons of equal area, extending the proven equilateral-triangle and square cases to general n.

Background

The paper identifies the Neumann counterpart of the polygonal spectral optimization problem: by analogy with classical isoperimetric inequalities, the regular n-gon is expected to maximize the first nonzero Laplace–Neumann eigenvalue among equal-area n-gons. The paper resolves the quadrilateral case by proving that the square is the unique maximizer, and it cites the equilateral-triangle case as previously established; the general polygonal problem is left as an unresolved conjectural direction.

References

By analogy with the Faber--Krahn and Szeg\H{o}--Weinberger inequalities, the regular $n$-gon is suspected to be a maximizer for the first nonzero Laplace--Neumann eigenvalue in the same class; for the maximization of higher Neumann eigenvalues among general planar domains, see.

— Maximizing the fundamental Laplace--Neumann eigenvalue on quadrilaterals  (2609.02784 - Endo et al., 2 Sep 2026) in Section 1, Introduction