A Pólya--Szegő Theorem for Tangential Polygons

We prove that, for every integer $N\ge3$, the regular $N$-gon uniquely maximizes torsional rigidity among all tangential $N$-gons of prescribed area. Since every triangle is tangential, the case $N=3$ yields an independent proof of the classical triangular P\'olya--Szeg\H{o} theorem. The proof decomposes a tangential polygon into mixed Dirichlet--Neumann right-triangular cells. Its analytic core is the strict concavity of $$\alpha\longmapsto h(\tan\alpha)-\frac18\tan\alpha,$$ where $h$ is the mixed torsional rigidity of a right-triangular cell with one Dirichlet side and two Neumann sides, and $\alpha\in(0,\pi/2)$ is the angle between the Neumann sides. We also obtain an explicit deficit decomposition that separates angular asymmetry from perimeter excess. As applications, we give a novel analytic proof that the torsional rigidity of equal-area regular polygons increases strictly with the number of sides, and derive an explicit criterion ensuring that a tangential polygon has larger first Dirichlet eigenvalue than the equal-area regular polygon.