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.
