For every integer n≥ 3 and every area A>0, among all convex n-gons P⊂R^2 with |P|=A, the regular n-gon R_n,A uniquely minimizes the first Dirichlet eigenvalue of the Laplacian: lambda_1(P)≥ lambda_1(R_n,A). Equality holds if and only if P is congruent to R_n,A, that is, differs from it by a rigid motion.

Pólya–Szegő conjecture for convex polygons

The Pólya–Szegő conjecture, posed by Pólya and Szegő in 1951, says that among fixed-area convex $n$ -gons, the regular $n$ -gon uniquely minimizes the first Dirichlet eigenvalue, up to rigid motions.

Known results

Triangles and quadrilaterals: global optimality is known.
$n=5,6$ : validated computation proves local minimality of the regular polygon (2024).
Earlier numerical and validated-computing work addressed further small values, but did not establish global optimality.

September 2026 large-side-number result

A September 16, 2026 report identifies Zhuo Cheng, Changfeng Gui, Yeyao Hu, and Qinfeng Li’s preprint The Pólya–Szegő conjecture for convex polygons with many sides. It claims that the regular polygon is the unique minimizer for all sufficiently large $N$ , using a Fourier inequality and spectral stability; the claim is unverified and does not cover every $N$ .

Current status (as of September 2026): the triangle and quadrilateral cases are settled, and a claimed but unverified result covers sufficiently large $N$ ; the remaining side numbers are open.

Pólya–Szegő conjecture for convex polygons

Primary source

Additional references

Progress summary

Solutions 0