For each integer N≥3, the Pólya–Szegö conjecture (1951), also called the polygonal Faber–Krahn inequality, asserts that among all plane polygons with at most N sides and a fixed area, the regular N-gon uniquely minimizes the first Dirichlet eigenvalue λ₁ of the Laplacian. Pólya proved the cases N=3 (equilateral triangle) and N=4 (square) by Steiner symmetrization, which cannot decide N≥5 because symmetrization can increase the number of sides. The conjecture is open for every N≥5. Modern partial results include a reduction of each case N≥5 to a finite (but astronomically large) set of certified numerical computations (Bogosel–Bucur), rigorous local minimality of the regular pentagon and hexagon via interval arithmetic (Bogosel–Bucur), a large-N manifold result (Indrei), and a 2026 computer-assisted proof that λ₁ of the regular N-gon is strictly decreasing in N — a necessary condition — which still leaves the global minimization open. The problem asks for a proof or refutation of the inequality for N≥5.

Background

For a bounded plane domain Ω, the Dirichlet eigenvalue problem asks for numbers λ and nonzero functions u with −Δu=λu-\Delta u=\lambda u in Ω and u=0u=0 on the boundary; the smallest such number λ1(Ω)>0\lambda_1(\Omega)>0 is the fundamental frequency (the square of the base vibrational frequency of a fixed membrane with the shape of Ω), and it is characterized variationally by the Rayleigh quotient λ1(Ω)=min⁡u≠0∫Ω∣∇u∣2 dx/∫Ωu2 dx\lambda_1(\Omega)=\min_{u\ne 0}\int_\Omega|\nabla u|^2\,dx/\int_\Omega u^2\,dx over functions vanishing on the boundary. λ₁ scales like the inverse of the area, so comparing domains of equal area is natural. The classical Faber–Krahn theorem (reversed isoperimetric inequality for λ₁) states that among all plane domains of a given area, the disk uniquely minimizes λ₁; its standard proof uses Steiner symmetrization, a rearrangement that replaces a domain by a mirror-symmetric one of the same area without increasing λ₁. Pólya and Szegö, in their 1951 book Isoperimetric Inequalities in Mathematical Physics, posed the analogous question with the disk replaced by regular polygons: among polygons with a given number of sides and fixed area, is the regular polygon the unique minimizer of λ₁? Pólya had proved this for N=3 (equilateral triangle) and N=4 (square) by Steiner symmetrization; since symmetrizing a polygon with five or more sides can produce extra sides, the method fails for N≥5, and the book even contemplated the possibility of counterexamples. The formulation with 'at most N sides' is equivalent to the one with 'exactly N sides', because a minimizer is known to exist in the class of polygons with at most N sides and to have exactly N sides. The conjecture, sometimes called the polygonal Faber–Krahn inequality, is Conjecture 2.3 (attributed to D. Bucur) in the AIM problem list on shape optimization, and the 2017 De Gruyter volume Shape Optimization and Spectral Theory surveys the surrounding theory. A weaker, necessary statement — that λ₁ of the regular N-gon of fixed area strictly decreases with N, so regular polygons approach the disk monotonically — was conjectured by Antunes–Freitas in 2006, recorded as Conjecture 6.10 in that 2017 volume and as Question 2.3.3 in the AIM list, and proved by a computer-assisted argument in 2026. Beyond its intrinsic interest in spectral geometry, the conjecture would give sharp eigenvalue bounds on polygonal domains, the shapes that arise universally as computational meshes in numerical analysis.

Problem Statement

For every integer N≥3 and every A>0, prove — or disprove — that among all plane polygons with at most N sides and area A, the regular N-gon RN,AR_{N,A} uniquely minimizes the first Dirichlet eigenvalue of the Laplacian: λ1(P)≥λ1(RN,A)\lambda_1(P)\ge\lambda_1(R_{N,A}) for every such polygon P, with equality if and only if P is the regular N-gon. Since the cases N=3 and N=4 are classical (Pólya), the open core is every N≥5; a complete answer establishes the inequality for all N≥3, and a resolution of any single fixed N≥5 settles that case of the conjecture (a refutation, by an explicit N-gon of equal area with strictly smaller λ₁ for some N≥5, would disprove the conjecture).

References

  1. G. Pólya and G. Szegö, Isoperimetric Inequalities in Mathematical Physics, Annals of Mathematics Studies no. 27, Princeton University Press, Princeton, N.J., 1951. DOI 10.1515/9781400882663, https://doi.org/10.1515/9781400882663 (zbMATH record Zbl 0044.38301, https://zbmath.org/0044.38301)
  2. Pedro Antunes and Pedro Freitas, New bounds for the principal Dirichlet eigenvalue of planar regions, Experimental Mathematics 15(3):333–342, 2006. DOI 10.1080/10586458.2006.10128966, https://doi.org/10.1080/10586458.2006.10128966
  3. Carlo Nitsch, On the first Dirichlet Laplacian eigenvalue of regular polygons, Kodai Mathematical Journal 37(3):595–607, 2014. DOI 10.2996/kmj/1414674611, https://doi.org/10.2996/kmj/1414674611
  4. Antoine Henrot (ed.), Shape optimization and spectral theory, De Gruyter Open, 2017. DOI 10.1515/9783110550887, https://doi.org/10.1515/9783110550887
  5. Beniamin Bogosel and Dorin Bucur, On the polygonal Faber-Krahn inequality, Journal de l'École polytechnique — Mathématiques 11:19–105, 2024. DOI 10.5802/jep.250, https://doi.org/10.5802/jep.250 (preprint arXiv:2203.16409)
  6. Emanuel Indrei, On the first eigenvalue of the Laplacian for polygons, Journal of Mathematical Physics 65(4):041506, 2024. DOI 10.1063/5.0179618, https://doi.org/10.1063/5.0179618 (preprint arXiv:2210.14806)
  7. David Berghaus, Bogdan Georgiev, Hartmut Monien, and Danylo Radchenko, On Dirichlet eigenvalues of regular polygons, Journal of Mathematical Analysis and Applications 538(2):128460, 2024. DOI 10.1016/j.jmaa.2024.128460, https://doi.org/10.1016/j.jmaa.2024.128460
  8. Beniamin Bogosel and Dorin Bucur, Polygonal Faber-Krahn inequality: Local minimality via validated computing, arXiv:2406.11575, 2024, https://arxiv.org/abs/2406.11575
  9. Joel Dahne, Javier Gómez-Serrano, and Joana Pech-Alberich, Monotonicity of the first Dirichlet eigenvalue of regular polygons, arXiv:2601.16285, 2026, https://arxiv.org/abs/2601.16285
  10. A. Henrot, E. Harrell, D. Bucur, and S. Steinerberger, AIM Problem Lists: Shape optimization with surface interactions — The Dirichlet Laplacian, Conjecture 2.3, http://aimpl.org/shapesurface/2/
  11. Changfeng Gui, Yeyao Hu, and Qinfeng Li, A Quantitative Pólya–Szegő Theorem for Tangential Polygons, arXiv:2607.28768, 2026, https://arxiv.org/abs/2607.28768
  12. Changfeng Gui, Yeyao Hu, Qinfeng Li, and Chenyang Zhang, Mixed Torsion on Right Triangles and the Pólya–Szegő Monotonicity Problem for Regular Polygons, arXiv:2606.13448, 2026, https://arxiv.org/abs/2606.13448