The well-known Faber-Krahn isoperimetric inequality states that among all domains with a given volume, the ball is the minimizer of the first Dirichlet-Laplacian eigenvalue. Therefore, it's natural to make the conjecture that among all polygonal domains with a given area and a given number of sides, the regular polygon minimizes the first eigenvalue. Pólya gave a proof for triangles and quadrilaterals using the technique of Steiner symmetrization, but for polygons with more than 4 sides, this remains an open problem. A review of Pólya's classical proof and of a numerical evaluation by Bogosel and Bucur is given. Furthermore, other approaches related to the eigenvalue optimization problem, including techniques from rearrangement theory, Cheeger theory, and curvature flow theory, are explored.