On the eigenvalues and the torsional rigidities of regular Reuleaux polygons

Abstract

Let $\lambda _1({\tilde{R}}_{2n+1})$ be the first Dirichlet eigenvalues of the Laplacian associated to regular Reuleaux polygons ${\tilde{R}}_{2n+1}$ of width ${\tilde{w}}_{2n+1}=1$, $n=1,2,3,\ldots$Let $T({\tilde{R}}_{2n+1})$ be their torsional rigidities. It is established that $\lambda _1({\tilde{R}}_{2n+1})$ is a decreasing sequence, and $T({\tilde{R}}_{2n+1})$ an increasing sequence. Similar (but reversed) inequalities hold in the class of regular Reuleaux polygons circumscribed to the unit circle.