Pólya conjecture for regular polygons

Establish that, among planar n-gons of prescribed equal area, the regular n-gon minimizes the first Laplace–Dirichlet eigenvalue for all n, beyond the cases n=3 and n=4 that have already been settled.

Background

The paper recalls the long-standing Pólya–Szegő conjecture concerning the first Laplace–Dirichlet eigenvalue among planar polygons with prescribed area. The conjecture asserts optimality of the regular polygon, while the authors note that only the triangular and quadrilateral cases had been settled in the cited literature at the time of writing. Thus the general n-gon problem remains unresolved for higher numbers of sides.

References

Pólya conjectured that the regular $n$-gon minimizes the first Laplace--Dirichlet eigenvalue among $n$-gons of prescribed area. Despite its apparent simplicity, the conjecture is notoriously difficult: only the cases $n = 3$ and $n = 4$ have been settled, both via Steiner symmetrization.

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