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.
