  The Faber-Krahn inequality in $\mathbb{R}^2$ states that among all open
bounded sets of given area the disk minimizes the first Dirichlet Laplacian
eigenvalue. There are numerical evidences that for all $N\ge 3$ the first
Dirichlet Laplacian eigenvalue of the regular $N$-gon is greater than the one
of the regular $(N+1)$-gon of same area. This natural property is also
suggested by the fact that the shape of regular polygons becomes more and more
"rounded" as $N$ increases and, among sets of given area, disk minimize the
eigenvalue. Aiming to settle such a conjecture, in this work we investigate
possible ways to estimate the difference between eigenvalues of regular
$N$-gons and $(N+1)$-gons.
