Year: 2026
Type: preprint
Abstract: For every sufficiently large N, we prove that the regular N-gon uniquely minimizes the first Dirichlet eigenvalue among convex polygons of prescribed area with at most N sides, up to rigid motions. Quantitative Faber--Krahn stability localizes minimizers near the disk but leaves their discrete geometry undetermined. We encode that geometry by a centered measure of exterior angles and prove an inverse-cubic Fourier inequality whose equality case is the uniform root configuration. Its defect gives a lower bound for the quadratic spectral excess over the regular polygon. Cyclic cancellation and a common material-coordinate comparison make the nonlinear difference small relative to the same defect. Exact constraint coordinates extend this comparison to arbitrarily small positive defects. Minimality then forces zero defect, which characterizes the regular polygon.
Source: arXiv (Cornell University)
Authors: Cheng, Zhuo, Gui, Changfeng, Hu, Yeyao, Li, Qinfeng
Institutions: Central South University, Hunan University
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Funder: Universidade de Macau
OpenAlex ID: W7213523160