The Reinhardt and Ulam Conjectures

1934年，Reinhardt考虑了在最密集的堆积密度最低的平面上确定中心对称凸盘形状的问题。在非正式用语中，如果合同要求守财奴用一盘相同的金币尽可能密集地装满托盘付款，如果合同规定硬币是凸起的和中心对称的，那么守财奴应该选择什么形状的硬币才能尽可能少地抛售黄金？莱因哈特推测，硬币的形状应该是光滑的八角形。平滑的八角形是通过取一个正八角形并用双曲线圆弧修剪角点来构建的。平滑后的八角形的密度约为90%。以前的研究人员对这一猜想的研究往往集中在特殊情况下。通过对PI的研究，对这一问题进行了总体分析。引入了一个关于二元特殊线性群的变分问题，该问题反映了Reinhardt猜想的结构。这个问题的一个有趣的特点是，猜想的解不是解析的，而只满足Lipschitz条件。这个问题的第二个值得注意的特征是存在一个有限个变量的非线性优化问题，将光滑多边形与猜想最优的光滑八边形联系起来。PI之前已经完成了许多与Reinhardt猜想证明相关的计算，并建议完成Reinhardt猜想的证明。这项研究将解决Reinhardt在1934年提出的关于平面上凸形的最优填充密度尽可能小的猜想。这项提议的意义体现在其更广泛的背景下。在这里，数学研究的三个重要领域涉及到一个问题：离散几何、非光滑变分分析和全局非线性最优化。有关堆积和密度的问题是离散几何的核心，与材料科学中经常出现的相同性质的问题密切相关。变分问题和更广泛的控制理论已经成为许多学科中不可或缺的工具，从数学金融到机器人控制。然而，给出精确非光滑解的研究相对较少，这一特点使本课题在变分问题中具有特殊的兴趣。这项研究还有望进一步发展使用计算机获得非线性优化问题的准确全局解的方法。在整个科学中，非线性最优化的应用非常广泛，当在一个参数有限的系统中寻找最佳选择时，非线性最优化就自然而然地出现了。因此，使用计算机寻找精确解的方法有可能得到广泛应用。因此，通过研究这种特殊的布局问题，数学工具可能会进一步发展，在整个科学中具有广泛的应用前景。

In 1934, Reinhardt considered the problem of determining the shape of the centrally symmetric convex disk in the plane whose densest packing has the lowest density. In informal terms, if a contract requires a miser to make payment with a tray of identical gold coins filling the tray as densely as possible, and if the contract stipulates the coins to be convex and centrally symmetric, then what shape of coin should the miser choose in order to part with as little gold as possible? Reinhardt conjectured that the shape of the coin should be a smoothed octagon. The smoothed octagon is constructed by taking a regular octagon and clipping the corners with hyperbolic arcs. The density of the smoothed octagon is approximately 90 per cent. Work by previous researchers on this conjecture has tended to focus on special cases. Research of the PI gives a general analysis of the problem. It introduces a variational problem on the special linear group in two variables that captures the structure of the Reinhardt conjecture. An interesting feature of this problem is that the conjectured solution is not analytic, but only satisfies a Lipschitz condition. A second noteworthy feature of this problem is the presence of a nonlinear optimization problem in a finite number of variables, relating smoothed polygons to the conjecturally optimal smoothed octagon. The PI has previously completed many calculations related to the proof of the Reinhardt conjecture and proposes to complete the proof of the Reinhardt conjecture.This research will solve a conjecture made in 1934 by Reinhardt about the convex shape in the plane whose optimal packing density is as small as possible. The significance of this proposal is found in its broader context. Here, three important fields of mathematical inquiry are brought to bear on a single problem: discrete geometry, nonsmooth variational analysis, and global nonlinear optimization. Problems concerning packings and density lie at the heart of discrete geometry and are closely connected with problems of the same nature that routinely arise in materials science. Variational problems and more generally control theory are have become indispensable tools in many disciplines, ranging from mathematical finance to robotic control. However, research that gives an exact nonsmooth solution is relatively rare, and this feature gives this project special interest among variational problem. This research is also expected to further develop methods that use computers to obtain exact global solutions to nonlinear optimization problems. Applications of nonlinear optimization are abundant throughout science and arise naturally whenever a best choice is sought among a system with finitely many parameters. Methods that use computers to find exact solutions thus have the potential of finding widespread use. Thus, by studying this particular packing problem, mathematical tools may be further developed with promising prospects of broad application throughout the sciences.