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A quantitative Robin Pólya--Szegő theorem for tangential polygons

  • Changfeng Gui
  • Yeyao Hu
  • Qinfeng Li
  • Hang Yang
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Abstract

We extend the quantitative Dirichlet theorem of Gui, Hu, and Li \cite{GuiHuLi2026} to Robin torsional rigidity. For every fixed β>0, we prove that the regular N-gon uniquely maximizes Robin torsional rigidity among tangential N-gons of prescribed area. Our explicit perimeter deficit bound contains the Dirichlet deficit obtained by Gui, Hu, and Li \cite{GuiHuLi2026} and an additional Robin term that is positive for nonregular tangential polygons. For triangles, we answer the fixed-area optimization problem recorded as open by Keady and Wiwatanapataphee \cite[Appendix A.1.2]{KeadyWiwatanapataphee2016}. We also establish strict monotonicity along equal-area regular polygons as the number of sides increases, with an explicit bound for each consecutive gap. The proof combines mixed Robin--Neumann triangular cells, spectral estimates for angular concavity, and a quantitative scaling estimate.

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Cite

BibTeXRIS
BibTeX
@misc{gui2026quantitative,
  title = {{A quantitative Robin Pólya--Szegő theorem for tangential polygons}},
  author = {Changfeng Gui and Yeyao Hu and Qinfeng Li and Hang Yang},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.21417},
  eprint = {2610.21417},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.21417},
}
RIS
TY  - UNPB
TI  - A quantitative Robin Pólya--Szegő theorem for tangential polygons
AU  - Changfeng Gui
AU  - Yeyao Hu
AU  - Qinfeng Li
AU  - Hang Yang
PY  - 2026
DA  - 2026/10/15
PB  - arXiv
DO  - 10.48550/arxiv.2610.21417
UR  - https://arxiv.org/abs/2610.21417
AB  - We extend the quantitative Dirichlet theorem of Gui, Hu, and Li \cite{GuiHuLi2026} to Robin torsional rigidity. For every fixed β>0, we prove that the regular N-gon uniquely maximizes Robin torsional rigidity among tangential N-gons of prescribed area. Our explicit perimeter deficit bound contains the Dirichlet deficit obtained by Gui, Hu, and Li \cite{GuiHuLi2026} and an additional Robin term that is positive for nonregular tangential polygons. For triangles, we answer the fixed-area optimization problem recorded as open by Keady and Wiwatanapataphee \cite[Appendix A.1.2]{KeadyWiwatanapataphee2016}. We also establish strict monotonicity along equal-area regular polygons as the number of sides increases, with an explicit bound for each consecutive gap. The proof combines mixed Robin--Neumann triangular cells, spectral estimates for angular concavity, and a quantitative scaling estimate.
ER  -