Preface

Summary

The year 1964 witnessed the publication of two fundamental monographs about infinite packing and covering: László Fejes Tóth's Regular Figures, which focused on arrangements in surfaces of constant curvature, and Claude Ambrose Rogers's Packing and Covering, which discussed translates of a given convex body in higher dimensional Euclidean spaces. This is the finite counterpart of the story told in these works. I discuss arrangements of congruent convex bodies that either form a packing in a convex container or cover a convex shape. In the spherical and the hyperbolic space I only consider packings and coverings by balls. The most frequent quantity to be optimized is the density, which is the ratio of total volume of the congruent bodies over the volume either of the container or of the shape that is covered. In addition, extremal values of the surface area, mean width, or other fundamental quantities are also investigated in the Euclidean case. A fascinating feature of finite packings and coverings is that optimal arrangements are often related to interesting geometric shapes.

The main body of the book consists of two parts, followed by the Appendix, which discusses some important background information and prerequisites. Part 1 collects results about planar arrangements. The story starts with Farkas Bolyai and Axel Thue, who investigated specific finite packings of unit discs in the nineteenth century. After a few sporadic results, the theory of packings and coverings by copies of a convex domain started to flourish following the work of László Fejes Tóth.