A Quantitative Pólya--Szegő Theorem for Tangential Polygons

Abstract

For a bounded Lipschitz domain $\Omega \subset {\mathbb{R}}^2$, let $T(\Omega )={\int }_{\Omega }{u}_{\Omega }dx$ denote its torsional rigidity, where $-\Delta u_{\Omega }=1$ in $\Omega$ and $u_{\Omega }=0$ on $\partial \Omega$. We prove a quantitative Pólya--Szegő inequality for tangential polygons. Let $N\ge 3$, let $P$ be a tangential $N$-gon, set $A=|P|$, and let $R_N$ be the regular $N$-gon of area $A$. Writing $L(\cdot )$ for perimeter, we obtain the explicit deficit estimate \[ T(R_N)-T(P)\ge \frac{A^2}{8N\tan(π/N)} \left(1-\frac{L(R_N)^2}{L(P)^2}\right)^2.\]Thus, at fixed area, the torsional deficit is controlled from below purely by the perimeter ratio. In particular, the regular $N$-gon is the unique maximizer of torsional rigidity among tangential $N$-gons of prescribed area; for $N=3$ this gives the classical triangular Pólya--Szegő theorem with a quantitative estimate. The same perimeter estimate yields an explicit positive lower bound for $T(R_{N+1})-T(R_N)$ for equal-area regular polygons, and hence a rather short alternative proof of the strict monotonicity of torsional rigidity in $N$. Combined with the Kohler--Jobin inequality, it also gives an explicit sufficient condition for the polygonal Faber--Krahn inequality within the tangential class. Our full quantitative inequality is stronger: it contains, in addition, a nonnegative angular Jensen deficit, which yields quantitative angular stability away from degenerate configurations.