On the optimization of the Robin eigenvalues in some classes of polygons

Given the eigenvalue problem for the Laplacian with Robin boundary conditions, (with $\beta \in \mathbb{R}\setminus \{0\}$ the Robin parameter), we consider a shape minimization problem for a function of the first eigenvalues if $\beta >0$ and a shape maximization problem if $\beta <0$. Both problems are settled in a suitable class of generalized polygons with an upper bound on the number of sides, under either perimeter or volume constraint.