Topics in Discrete Geometry: Packing and Covering

两位研究者建议继续他们的研究包装和覆盖空间与凸体的一致副本，包括主题的分析和组合方面。在他们之前的工作中，他们得到了很多二维的结果，也得到了一些三维或更高维度的结果。建议的研究集中在3维的覆盖问题上，并且在可行的情况下，在更高的维度上进行覆盖。连接各种问题的主线是三维球面覆盖猜想。本文拟研究的主要课题是：用平行球串覆盖空间；用无限长的等宽圆柱覆盖空间；（平面的）有边缘的圆形覆盖物；球形覆盖物的稳定性。上面提到的主题属于离散几何领域，其中一些主题，特别是那些处理格排列的主题，与数字几何有关。研究者解决的问题有一种直观的味道和一种自然的动机。例如，球面覆盖问题可以解释为寻找覆盖整个空间的范围为球形且大小相等的发射站的最经济分布，即使每个点都在至少一个发射站的可及范围内。每当发现图形或实体排列的某种最优性条件（例如，最小密度，如在输变电站分布的例子中）意味着排列的某种规律性时，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.