平滑多边形的堆积
Packings of Smoothed Polygons
AI总结：
本书将凸圆盘最密堆积问题表述为 Lie 群上的最优控制问题，通过分析 Pontryagin 极值与抖振解，证明 Mahler 第一猜想：最难堆积的中心对称凸圆盘是平滑多边形。
摘要：
本书运用最优控制理论证明，平面中最难堆积的中心对称凸圆盘是一个平滑多边形。所谓平滑多边形，是指其角以一种特殊方式被双曲线弧修圆的多边形。高度难以堆积，意味着即使是该圆盘的最密堆积，其密度也很低。受 Minkowski 数的几何启发，研究人员从20世纪20年代初开始寻找最难堆积的中心对称凸圆盘（简称最难堆积圆盘）。1934年，Reinhardt 猜想最难堆积的圆盘是一个平滑八边形。Mahler 独立于 Reinhardt 开展工作，于1947年试图证明最难堆积的圆盘必定是平滑多边形，但未获成功。本书证明了 Mahler 当年想要证明的结论：关于平滑多边形的 Mahler 第一猜想。他的第二猜想与 Reinhardt 猜想相同，至今仍未解决。本书探讨了该堆积问题的许多显著结构，将其表述为 Lie 群上的一个最优控制理论问题，并与双曲几何和 Hamiltonian 力学相联系。该最优控制问题的 Bang-bang Pontryagin 极值正是平滑多边形。证明中出现了极大困难，原因在于最优控制问题中的抖振行为，它对应于可能存在的、具有无穷多条边的平滑多边形，而这些情形必须被排除。为分析并消除抖振解的可能性，本书引入了一个离散动力系统（Poincaré 首次回归映射），并在以奇异集为中心的爆破上完整描述了其不动点、稳定与不稳定流形以及吸引盆。本书中的部分证明借助计算机代数系统以计算机辅助方式完成。
英文摘要：
This book uses optimal control theory to prove that the most unpackable centrally symmetric convex disk in the plane is a smoothed polygon. A smoothed polygon is a polygon whose corners have been rounded in a special way by arcs of hyperbolas. To be highly unpackable means that even densest packing of that disk has low density. Motivated by Minkowski's geometry of numbers, researchers began to search for the most unpackable centrally symmetric convex disk (in brief, the most unpackable disk) starting in the early 1920s. In 1934, Reinhardt conjectured that the most unpackable disk is a smoothed octagon. Working independently of Reinhardt, Mahler attempted without success in 1947 to prove that the most unpackable disk must be a smoothed polygon. This book proves what Mahler set out to prove: Mahler's First conjecture on smoothed polygons. His second conjecture is identical to the Reinhardt conjecture, which remains open. This book explores the many remarkable structures of this packing problem, formulated as a problem in optimal control theory on a Lie group, with connections to hyperbolic geometry and Hamiltonian mechanics. Bang-bang Pontryagin extremals to the optimal control problem are smoothed polygons. Extreme difficulties arise in the proof because of chattering behavior in the optimal control problem, corresponding to possible smoothed polygons with infinitely many sides that need to be ruled out. To analyze and eliminate the possibility of chattering solutions, the book introduces a discrete dynamical system (the Poincare first recurrence map) and gives a full description of its fixed points, stable and unstable manifolds, and basin of attraction on a blowup centered at a singular set. Some proofs in this book are computer-assisted using a computer algebra system.
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