最大面积小多边形问题
Maximum-Area Small Polygons of Even Order
AI总结：
该研究解决了直径不超过1的最大面积小多边形的缺失偶数阶问题，证明n≥8时最大面积小n边形的构型，结合已有结果得到n≥3时所有情况的精确解。
AI中文摘要：
小多边形是直径不超过1的平面多边形，最大面积问题是经典问题：奇数阶的情况已由Reinhardt解决，而偶数阶的情况无法通过正则性解决，且除低维情况外仍未解决。我们解决了缺失的偶数阶问题：对于每个偶数n≥8，每个面积最大的小n边形的直径图由一个(n-1)-环和一条悬挂边组成。我们将该构型转化为带标记的鲁洛多边形，通过精确包络论证消除悬挂顶点，为剩余的单位距离环推导离散Noether系统；Noether变量属于明确的开凸域，且满足生成作用Φ_{n-1}的临界点方程。主要分析步骤证明，该作用的Hessian在整个自然域上是负定的，两次精确因式分解将局部计算简化为显式正项，因此该作用有唯一临界点；重构后表明，最大化多边形在欧几里得等距和反射下是唯一的。结合经典奇数阶定理及已知的n=4、6的情况，得到了n≥3时所有情况的精确解。全局性是解析性的，文章附带的符号计算仅用于检查代数恒等式，不作为计算机辅助证明。
英文摘要：
A small n-gon is a planar n-gon whose diameter is at most one. For odd n, Reinhardt proved that the regular polygon is optimal. For even n, the maximizer is nonregular, and only a few low orders were known exactly. We prove that for every even n >= 8 the maximum-area small n-gon is unique up to Euclidean isometry and reflection. Foster and Szabo's description of the diameter graph reduces a maximizer to an (n-1)-cycle of unit distances together with one pendant diameter. We determine the compatible boundary order, interpret the cycle as the centers of a Reuleaux (n-1)-gon, and eliminate the pendant vertex by a one-variable area calculation. The remaining first-order equations have conserved translation and rotation quantities. In the resulting radial variables they become the critical-point equations of an explicit function on a convex domain. We prove that this function is strictly concave by factoring the relevant principal minors of its local Hessian. Compactness gives existence, while strict concavity gives uniqueness. The proof is analytic. The symbolic scripts supplied with the paper check algebraic identities but are not used as part of the proof. Combined with the classical odd-order and low-order results, this determines the maximal area for every n >= 3.
