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

Given the eigenvalue problem for the Laplacian with Robin boundary conditions, (with $\beta\in\R\setminus\{0\}$ the Robin parameter), we consider shape minimization problems if $\beta>0$ and shape maximization problems 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.