Topics in Discrete Geometry: Packing and Covering

两位研究人员建议继续研究凸体的全等复制品填充和覆盖空间的问题，包括这些主题的分析和组合方面。在他们以前的工作中，他们在维度2上得到了很多结果，在维度3和更高的维度上也得到了一些结果。建议的研究集中在维度3和更高维度的覆盖问题，只要可行的话。连接各种问题的共同主线是三维球面覆盖猜想。本文的主要研究内容包括：球面平行串覆盖空间；无限长同余圆柱覆盖空间；(平面)带边圆覆盖；球面覆盖的稳定性性质。上述主题属于离散几何领域，其中一些主题，特别是与格子排列有关的主题，与数的几何有关。调查人员解决的问题有一种直观的味道和自然的动机。例如，球面覆盖问题可以解释为搜索范围为球形且大小相等以覆盖整个空间的发射站的最经济分布，即使得每个点都在至少一个站的到达范围内。当发现图形或实体排列的某种最优化条件(例如，在传输站分布的例子中的最小密度)隐含着排列的某些规律性时，就为计算几何提供了一个强大的工具。这类结果为寻找高效的计算算法提供了理论基础。研究人员建议增加他们对这一重要而困难的研究领域的贡献。

The two investigators propose to continue their study of packing and covering of space with congruent replicas of a convex body, including the analytical and the combinatorial aspects of the topics. In their previous work, they obtained many results in dimension 2, and some in dimension 3 and higher. The proposed research concentrates on the covering problems in dimensions 3 and, whenever feasible, in higher dimensions. The common thread linking the various problems is the sphere covering conjecture in 3 dimensions. The following topics are the main subjects of the proposed investigation: covering space with parallel strings of spheres; covering space with congruent circular cylinders of infinite length; circle covering (of the plane) with a margin; stability properties of sphere coverings. The topics mentioned above belong to the area of discrete geometry, and some of them, especially those dealing with lattice arrangements, are related to the geometry of numbers. The problems addressed by the investigators have an intuitive flavor and a natural motivation. For instance, the sphere covering problem can be interpreted as searching for the most economical distribution of transmission stations whose ranges are spherical and of equal sizes to cover the whole space, i.e. so that every point is within reach of at least one of the stations. Whenever it is discovered that a certain optimality condition (e.g. minimum density, as in the example of transmission stations distribution) of an arrangement of figures or solids implies some regularity of the arrangement, a powerful tool is provided for computational geometry. Results of this type give a theoretical foundation for finding efficient computational algorithms. The investigators propose to increase their contributions to this important and difficult area of research.