Classical Topics in Discrete Geometry

Geometry is a classical core part of mathematics which, with its birth, marked the beginning of the mathematical sciences. Thus, not surprisingly, geometry has played a key role in many important developments of mathematics in the past, as well as in present times. While focusing on modern mathematics, one has to emphasize the increasing role of discrete mathematics, or equivalently, the broad movement to establish discrete analogues of major components of mathematics. In this way, the works of a number of outstanding mathema- cians including H. S. M. Coxeter (Canada), C. A. Rogers (United Kingdom), and L. Fejes-T oth (Hungary) led to the new and fast developing eld called discrete geometry. One can brie y describe this branch of geometry as the study of discrete arrangements of geometric objects in Euclidean, as well as in non-Euclidean spaces. This, as a classical core part, also includes the theory of polytopes and tilings in addition to the theory of packing and covering. D- crete geometry is driven by problems often featuring a very clear visual and applied character. The solutions use a variety of methods of modern mat- matics, including convex and combinatorial geometry, coding theory, calculus of variations, di erential geometry, group theory, and topology, as well as geometric analysis and number theory.

This monograph leads the reader to the frontiers of the very latest research developments in what is regarded as the central zone of discrete geometry. It is constructed around four classic problems in the subject, including the Kneser-Poulsen Conjecture.

A valuable source of geometric problems

Classical Topics Revisited.- Sphere Packings.- Finite Packings by Translates of Convex Bodies.- Coverings by Homothetic Bodies - Illumination and Related Topics.- Coverings by Planks and Cylinders.- On the Volume of Finite Arrangements of Spheres.- Ball-Polyhedra as Intersections of Congruent Balls.- Selected Proofs.- Selected Proofs on Sphere Packings.- Selected Proofs on Finite Packings of Translates of Convex Bodies.- Selected Proofs on Illumination and Related Topics.- Selected Proofs on Coverings by Planks and Cylinders.- Selected Proofs on the Kneser-Poulsen Conjecture.- Selected Proofs on Ball-Polyhedra.