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PreprintarXiv licencearXiv

Proof of the local version of the Pólya--Szegö conjecture for the torsional rigidity of polygons

  • Beniamin Bogosel
  • Dorin Bucur
  • Ilaria Fragalà
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Abstract

We prove the local version of the Pólya--Szegö conjecture for the torsional rigidity of polygons: for every n≥5, the regular n-gon is a strict local maximizer of torsional rigidity among convex n-gons of prescribed area. Our proof is entirely analytic. It builds on a locally stable proportional triangular covering inspired by Solynin and Zalgaller and an associated weighted Voronoi-type partition, together with a quantitative asymptotic analysis of the loss produced by truncating the overlapping triangles to the partition cells. The result is then obtained by establishing two key ingredients: the optimality of isosceles triangles for mixed torsional rigidity at fixed area and vertex angle, and a strict concavity property of the mixed torsional rigidity of isosceles triangles.

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Cite

BibTeXRIS
BibTeX
@misc{bogosel2026proof,
  title = {{Proof of the local version of the Pólya--Szegö conjecture for the torsional rigidity of polygons}},
  author = {Beniamin Bogosel and Dorin Bucur and Ilaria Fragalà},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.23051},
  eprint = {2609.23051},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.23051},
}
RIS
TY  - UNPB
TI  - Proof of the local version of the Pólya--Szegö conjecture for the torsional rigidity of polygons
AU  - Beniamin Bogosel
AU  - Dorin Bucur
AU  - Ilaria Fragalà
PY  - 2026
DA  - 2026/09/19
PB  - arXiv
DO  - 10.48550/arxiv.2609.23051
UR  - https://arxiv.org/abs/2609.23051
AB  - We prove the local version of the Pólya--Szegö conjecture for the torsional rigidity of polygons: for every n≥5, the regular n-gon is a strict local maximizer of torsional rigidity among convex n-gons of prescribed area. Our proof is entirely analytic. It builds on a locally stable proportional triangular covering inspired by Solynin and Zalgaller and an associated weighted Voronoi-type partition, together with a quantitative asymptotic analysis of the loss produced by truncating the overlapping triangles to the partition cells. The result is then obtained by establishing two key ingredients: the optimality of isosceles triangles for mixed torsional rigidity at fixed area and vertex angle, and a strict concavity property of the mixed torsional rigidity of isosceles triangles.
ER  -