Computer-assisted local maximality of regular polygons for torsional rigidity
Abstract
We study torsional rigidity as a function of the labeled vertices of a convex polygon. Starting from the distributed second shape derivative, we derive the Hessian with respect to vertex coordinates. At a regular polygon, dihedral symmetry makes this matrix block circulant in radial-tangential coordinates, reducing its spectrum to the eigenvalues of Hermitian matrices of order two. We also derive an exact second-variation Galerkin identity and guaranteed functional residual majorants. Finite elements approximate the PDE solutions entering the Hessian, and FLINT/Arb provides the interval arithmetic needed for certification. In the scale-invariant setting, we certify exactly four zero eigenvalues generated by similarities and 2n−4 strictly negative eigenvalues for 5≤ n≤25. The regular polygons in this range are therefore strict local maximizers, modulo similarities, of torsional rigidity divided by area squared. We prove broken regularity of the first and second material derivatives at each fixed regular polygon, and deduce an O(h2) error bound for the exact Galerkin Hessian and its eigenvalues on meshes fitted to the coarse fan.
Related
- Proof of the local version of the Pólya--Szegö conjecture for the torsional rigidity of polygons
- A sharp lower bound for torsional rigidity of convex domains
- Sharp Bounds on Torsional Rigidity of Metric Graphs with Signed Delta-Vertex Strengths
- Asymptotic analysis of Dirichlet problems in supercritically perforated domains via quantification by torsional rigidity
- Semiprime Rigidity and Growth along Powers in Integer-Sided Cyclic Polygons
BibTeX
@misc{bogosel2026computerassisted,
title = {{Computer-assisted local maximality of regular polygons for torsional rigidity}},
author = {Beniamin Bogosel},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.14785},
eprint = {2609.14785},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.14785},
}
RIS
TY - UNPB
TI - Computer-assisted local maximality of regular polygons for torsional rigidity
AU - Beniamin Bogosel
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2609.14785
UR - https://arxiv.org/abs/2609.14785
AB - We study torsional rigidity as a function of the labeled vertices of a convex polygon. Starting from the distributed second shape derivative, we derive the Hessian with respect to vertex coordinates. At a regular polygon, dihedral symmetry makes this matrix block circulant in radial-tangential coordinates, reducing its spectrum to the eigenvalues of Hermitian matrices of order two. We also derive an exact second-variation Galerkin identity and guaranteed functional residual majorants. Finite elements approximate the PDE solutions entering the Hessian, and FLINT/Arb provides the interval arithmetic needed for certification. In the scale-invariant setting, we certify exactly four zero eigenvalues generated by similarities and 2n−4 strictly negative eigenvalues for 5≤ n≤25. The regular polygons in this range are therefore strict local maximizers, modulo similarities, of torsional rigidity divided by area squared. We prove broken regularity of the first and second material derivatives at each fixed regular polygon, and deduce an O(h^2) error bound for the exact Galerkin Hessian and its eigenvalues on meshes fitted to the coarse fan.
ER -