The Reinhardt and Ulam Conjectures

1934年，Reinhardt考虑了确定平面上中心对称凸圆盘的形状的问题，该圆盘的密排密度最低。 用非正式的术语来说，如果一份合同要求守财奴用一盘相同的金币来支付，并且金币要尽可能地密集，如果合同规定金币是凸形的，并且是中心对称的，那么守财奴应该选择什么形状的金币才能尽可能少地拿走黄金呢？ 莱因哈特建议硬币的形状应该是平滑的八角形。 平滑的八边形是通过取一个正八边形并用双曲弧剪裁角来构造的。 光滑八边形的密度约为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。对PI的研究给出了问题的一般分析。 它介绍了一个变分问题的特殊线性群在两个变量，捕捉结构的莱因哈特猜想。 这个问题的一个有趣的特点是，约束解不是解析的，而是仅满足Lipschitz条件。 这个问题的第二个值得注意的特点是存在一个非线性优化问题，在一个有限数量的变量，相关的平滑多边形的最佳平滑八角形。 PI此前已经完成了与莱因哈特猜想证明相关的多项计算，并提出要完成莱因哈特猜想的证明，本次研究将解决莱因哈特在1934年提出的关于最佳堆积密度尽可能小的平面内凸形的猜想。这一建议的意义在于其更广泛的背景。 在这里，三个重要领域的数学调查带来承担一个问题：离散几何，非光滑变分分析，和全球非线性优化。 关于填充和密度的问题是离散几何的核心，并且与材料科学中经常出现的相同性质的问题密切相关。 变分问题和更一般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，这一特征使本项目在变分问题中特别感兴趣。 这项研究也有望进一步发展使用计算机获得非线性优化问题的精确全局解的方法。 非线性优化的应用在整个科学中是丰富的，并且每当在具有许多参数的系统中寻求最佳选择时自然出现。 因此，使用计算机找到精确解的方法具有广泛应用的潜力。 因此，通过研究这个特殊的包装问题，数学工具可能会进一步发展，在整个科学领域有着广阔的应用前景。

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.