In 1947, P\’olya proved that if $n=3,4$ the regular polygon $P_n$ minimizes

the principal frequency of an n-gon with given area $\alpha>0$ and suggested

that the same holds when $n \ge 5$. In $1951,$ P\’olya & Szeg\”o discussed the

possibility of counterexamples in the book “Isoperimetric Inequalities In

Mathematical Physics.” This paper constructs explicit $(2n-4)$–dimensional

polygonal manifolds $\mathcal{M}(n, \alpha)$ and proves the existence of a

computable $N \ge 5$ such that for all $n \ge N$, the admissible $n$-gons are

given via $\mathcal{M}(n, \alpha)$ and there exists an explicit set $

\mathcal{A}_{n}(\alpha) \subset \mathcal{M}(n,\alpha)$ such that $P_n$ has the

smallest principal frequency among $n$-gons in $\mathcal{A}_{n}(\alpha)$.

Inter-alia when $n \ge 3$, a formula is proved for the principal frequency of a

convex $P \in \mathcal{M}(n,\alpha)$ in terms of an equilateral $n$-gon with

the same area; and, the set of equilateral polygons is proved to be an

$(n-3)$–dimensional submanifold of the $(2n-4)$–dimensional manifold

$\mathcal{M}(n,\alpha)$ near $P_n$. If $n=3$, the formula completely addresses

a 2006 conjecture of Antunes and Freitas and another problem mentioned in

“Isoperimetric Inequalities In Mathematical Physics.” Moreover, a solution to

the sharp polygonal Faber-Krahn stability problem for triangles is given and

with an explicit constant. The techniques involve a partial symmetrization,

tensor calculus, the spectral theory of circulant matrices, and $W^{2,p}/BMO$

estimates. Last, an application is given in the context of electron bubbles.