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The publication of the first edition of Lagerungen in der Ebene, auf der Kugel und im Raum in 1953 marked the birth of discrete geometry. Since then, the book has had a profound and lasting influence on the development of the field. It included many open problems and conjectures, often accompanied by suggestions for their resolution. A good number of new results were surveyed by László Fejes Tóth in his Notes to the 2nd edition.
The present version of Lagerungen makes this classic monograph available in English for the first time, with updated Notes, completed by extensive surveys of the state of the art. More precisely, this book consists of:
- a corrected English translation of the original Lagerungen,
- the revised and updated Notes on the original text,
- eight self-contained chapters surveying additional topics in detail.
The English edition provides a comprehensive update to an enduring classic. Combining the lucid exposition of the original text with extensive new material, it will be a valuable resource for researchers in discrete geometry for decades to come.
List of contents
Part I. Lagerungen - Arrangements in the Plane, on the Sphere, and in Space.- 1. Some Theorems from Elementary Geometry.- 2. Theorems from the Theory of Convex Bodies.- 3. Problems on Packing and Covering in the Plane.- 4. Efficiency of Packings and Coverings with a Sequence of Convex Disks.- 5. Extremal Properties of Regular Polyhedra.- 6. Irregular Packing on the Sphere.- 5. Packing in Space.- Part II. Notes and Additional Chapters to the English Edition.- 8. Notes.- 9. Finite Variations on the Isoperimetric Problem.- 10. Higher Dimensions.- 11. Ball Packings in Hyperbolic Space.- 12. Mutliple Arrangements.- 13. Neighbors.- 14. Packing and Covering Properties fo Sequences of Convex Bodies.- 15. Four Classic Problems.- 16. Miscellaneous Problems about Packing and Covering.- References for Part I.- References for Part II.- Name Index.- Subject Index.
Report
"This book is both a very appealing English translation and an updated extension ... . The book ends with a comprehensive list of references, covering about 80 pages. ... Experts in the eld will appreciate it for its currentness and the vast bibliography. People new to the area will find an inspiring introduction and overview to the theory of arrangements in discrete geometry, as well as many invitations to participate in future developments." (Matthias Schymura, Mathematical Reviews, March, 2025)
