Pólya–Szegő polygonal Faber–Krahn conjecture
Fix an integer $n \ge 5$ and an area $A > 0$. For a simple planar polygon $P$ with exactly $n$ sides and area $A$, which need not be convex, let $\lambda_1(P)$ be the first Dirichlet eigenvalue of $-\Delta$ on $P$, and let $R_n$ be the regular $n$-gon of area $A$. Pólya and Szegő proved the inequality below for $n = 3$ and $n = 4$ by Steiner symmetrization, which can increase the number of sides when $n \ge 5$. Bogosel and Bucur reduced the problem, for each $n \ge 5$, to finitely many certified numerical computations, and proved by computations in interval arithmetic that $R_n$ is a local minimizer of $\lambda_1$ among $n$-gons of area $A$ for $n = 5, 6$. Cheng, Gui, Hu and Li (2026) proved that for every sufficiently large $n$, $R_n$ minimizes $\lambda_1$ among convex polygons of area $A$ with at most $n$ sides, uniquely up to rigid motions.
Problem. Is $\lambda_1(P) \ge \lambda_1(R_n)$ for every $n \ge 5$, every $A > 0$ and every such $P$, with equality only when $P$ is congruent to $R_n$?
Source: Pólya and Szegő (1951), who proved it for triangles and quadrilaterals; see Bogosel and Bucur (2024) and Levitin, Mangoubi and Polterovich (2023), Conjecture 5.1.19. Stated by AIM — Open Applied Problems for n ≥ 5.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- SP — Spectral Theory
- Secondary
- AP, OC
- Keywords
- AIM Open Applied Problems, Dirichlet eigenvalues, Faber–Krahn inequality, polygons
References
@book{polya1951,
author = {P{\'o}lya, George and Szeg{\H{o}}, Gabor},
title = {Isoperimetric Inequalities in Mathematical Physics},
series = {Annals of Mathematics Studies},
volume = {27},
publisher = {Princeton University Press},
address = {Princeton, NJ},
year = {1951},
doi = {10.1515/9781400882663},
}
@article{bogosel2024,
author = {Bogosel, Beniamin and Bucur, Dorin},
title = {On the polygonal {F}aber--{K}rahn inequality},
journal = {J. \'Ec. polytech. Math.},
volume = {11},
year = {2024},
pages = {19--105},
doi = {10.5802/jep.250},
eprint = {2203.16409},
archivePrefix = {arXiv},
}
@article{bogosel2026,
author = {Bogosel, Beniamin and Bucur, Dorin},
title = {Polygonal {F}aber--{K}rahn inequality: local minimality via validated computing},
journal = {Commun. Contemp. Math.},
year = {2026},
pages = {2650074},
doi = {10.1142/S0219199726500744},
eprint = {2406.11575},
archivePrefix = {arXiv},
}
@misc{cheng2026polygons,
author = {Cheng, Zhuo and Gui, Changfeng and Hu, Yeyao and Li, Qinfeng},
title = {The {P}{\'o}lya--{S}zeg{\H{o}} conjecture for convex polygons with many sides},
year = {2026},
eprint = {2609.18500},
archivePrefix = {arXiv},
}
@book{levitin2023,
author = {Levitin, Michael and Mangoubi, Dan and Polterovich, Iosif},
title = {Topics in Spectral Geometry},
series = {Graduate Studies in Mathematics},
volume = {237},
publisher = {American Mathematical Society},
address = {Providence, RI},
year = {2023},
doi = {10.1090/gsm/237},
}
@misc{aim-polygonal-faber-krahn,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Pólya–Szegő polygonal Faber–Krahn conjecture}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/a153f469ee0ea64e876b2ad97c79d252c19a3e22/problems/004-polygonal-faber-krahn.md},
note = {Entry 004 at commit a153f46, 2026-10-13},
}
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00296, "Pólya–Szegő polygonal Faber–Krahn conjecture", clopen.solutions.