最大面积小多边形的边界层与尖锐渐近分析
Boundary Layers and Sharp Asymptotics for Maximum-Area Small Polygons
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中文总结 AI 辅助
该研究确定直径不超过1的平面多边形最大面积$A_n$的渐近行为，通过分析偶阶最大化子的边界层与收敛性质，给出其渐近展开式及相关常数的精确变分定义。
中文摘要 AI 辅助
小多边形是直径至多为1的平面多边形，记$A_n$为阶数$n$对应的最大面积。利用姊妹论文中建立的偶阶最大化子的全局刻画，我们确定其渐近几何。对唯一悬垂直径附近的角亏进行缩放后，精确临界方程收敛到自治二阶递推关系，其标记边界条件选取唯一正半轴轨道，等价于显式严格凸作用量的唯一极小值。该轨道以稳定乘子$(-3+\boldsymbol{\text{√}}5)/2$趋近正则态，产生交替、指数衰减的边界层。一致有限周期跟踪定理将此剖面转移到真实最大化子。对每个固定深度，偏移裁剪论证证明，正变分有限截面是Bingane–Mossinghoff构造的极限几何域上的唯一全局极小值；该结论与在更宽松代数盒上的极小化明显不同。截面尖锐收敛，两步误差比为$|(-3+\boldsymbol{\text{√}}5)/2|^4$。极限常数$q_*$有精确变分定义和已验证的有理包围。当偶数$n\to\boldsymbol{\text{∞}}$时，$A_n=\frac{\boldsymbol{\text{π}}}{4}-\frac{5\boldsymbol{\text{π}}^3}{48n^2}-\frac{q_*\boldsymbol{\text{π}}^3}{n^3}+O(n^{-4})$。我们还确定了Foster–Szabo上界的领先间隙，并证明$A_n$可由$1/n$的实解析函数表示，误差为指数小量。
英文摘要
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 $n\to\infty$, $A_n=\frac\pi4-\frac{5π^3}{48n^2}-\frac{q_*π^3}{n^3}+O(n^{-4}).$ 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 $1/n$.
