Boundary Layers and Sharp Asymptotics for Maximum-Area Small Polygons

Abstract

A small polygon is a planar polygon of diameter at most one; let $A_n$ be the largest area at order $n$. Using the global characterization of the even-order maximizers established in a companion paper, we determine their asymptotic geometry. After scaling the angular deficits near the unique pendant diameter, the exact critical equations converge to an autonomous second-order recurrence. Its marked boundary condition selects a unique positive half-line orbit, equivalently the unique minimizer of an explicit strictly convex action. The orbit approaches the regular state with stable multiplier $(-3+\sqrt5)/2$, producing an alternating, exponentially damped boundary layer. A uniform finite-cycle shadowing theorem transfers this profile to the true maximizers. For every fixed depth, an excursion-clipping argument proves that the positive variational finite section is the unique global minimizer on the limiting geometric domain of the Bingane--Mossinghoff construction; this conclusion is expressly distinct from minimization on a looser algebraic box. The sections converge sharply, with two-step error ratio $|(-3+\sqrt5)/2|^4$. The limiting constant $q_*$ has an exact variational definition and a certified rational enclosure. For even $n\to\infty$, $A_n=\frac\pi4-\frac{5\pi^3}{48n^2}-\frac{q_*\pi^3}{n^3}+O(n^{-4}).$ We also identify the leading gap from the Foster--Szabo upper bound and prove that $A_n$ is represented, up to an exponentially small error, by a real-analytic function of $1/n$.