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.
Author(s): László Fejes Tóth, Gábor Fejes Tóth, Włodzimierz Kuperberg
Series: Grundlehren der mathematischen Wissenschaften, 360
Publisher: Springer
Year: 2023
Language: English
Pages: 453
City: London
Foreword
Preface to the English Edition
Preface to the Second Edition
Preface to the First Edition
Contents
Part I Lagerungen – Arrangements in the Plane, on the Sphere, and in Space
Chapter 1 Some Theorems from Elementary Geometry
1.1 Convex Sets
1.2 Affinity and Polarity
1.3 Extremum Properties of the Regular Polygons
1.4 The Isoperimetric Problem
1.5 Some Inequalities on Triangles
1.6 Euler’s Theorem on Polyhedra
1.7 The Regular and Semiregular Polyhedra
1.8 Polar Triangles, Lexell’s Circle
1.9 Some Identities in Vector Algebra
1.10 Some Formulae of Spherical Trigonometry
1.11 Historical Remarks
Chapter 2 Theorems from the Theory of Convex Bodies
2.1 Blaschke’s Selection Theorem
2.2 Jensen’s Inequality
2.3 Dowker’s Theorems
2.4 An Extremum Property of the Ellipse
2.5 On the Affine Perimeter
2.6 Variational Problems Regarding Affine Length
2.7 Rudiments of Integral Geometry
2.8 Historical Remarks
Chapter 3 Problems on Packing and Covering in the Plane
3.1 Density of Arrangements of Domains
3.2 The Problems of Densest Packing and Thinnest Covering with Circles
3.3 Some Outlines of Proofs
3.4 Packing and Covering Convex Disks with Congruent Circles
3.5 Dissecting a Convex Domain into Convex Parts
3.6 Packing a Convex Domain with Circles of ? Different Sizes
3.7 Estimates for Incongruent Circles
3.8 A Further Theorem on Covering with Circles
3.9 Dissecting a Convex Hexagon into Convex Polygons
3.10 Packing and Covering a Convex Hexagon with Congruent Convex Disks
3.11 A Packing Problem with Respect to Affine Length
3.12 On a Mean Value Formula
3.13 Historical Remarks
Chapter 4 Efficiency of Packings and Coverings with a Sequence of Convex Disks
4.1 Extremum Properties of Triangles
4.2 Centrally Symmetric Domains
4.3 Packing and Covering Efficiency of Sequences of Disks
4.4 Covering with Fragmented Disks
4.5 Historical Remarks
Chapter 5 Extremal Properties of Regular Polyhedra
5.1 Packing and Covering the Sphere with Congruent Spherical Caps
5.2 Some Additional Proofs
5.3 Approximating a Ball by Polyhedra
5.4 Volume of a Circumscribed Polyhedron
5.5 Volume of an Inscribed Polyhedron
5.6 Inequalities Linking the Inradius and Circumradius of Polyhedra
5.7 Isoperimetric Problems for Polyhedra
5.8 A General Inequality
5.9 On the Shortest Net Dissecting the Sphere into Convex Parts of Equal Area
5.10 On the Total Length of the Edges of a Polyhedron
5.11 The Thinnest Saturated Packing of Spherical Caps
5.12 Approximating a Convex Surface by Polyhedra
5.13 Historical Remarks
Chapter 6 Irregular Packing on the Sphere
6.1 The Graph Associated with a Family of Points
6.2 The Maximal Configuration for ? = 7
6.3 The Maximal Configuration for ? = 8 and 9
6.4 Some Configurations of More Than 9 Points
6.5 A Survey Table
6.6 Historical Remarks
Chapter 7 Packing in Space
7.1 General Remarks
7.2 The Problem of Densest Ball Packing
7.3 On an Extremal Space Partition
7.4 The Mean Value Formula in Space
7.5 Historical Remarks
Part II Notes and Additional Chapters to the English Edition
Chapter 8 Notes
8.1 Notes on Chapter 1
8.1.1 Notes on Section 1.2
8.1.2 Notes on Section 1.3
8.1.3 Notes on Section 1.4
8.1.4 Notes on Section 1.5
8.1.5 Notes on Section 1.6
8.1.6 Notes on Section 1.7
8.1.7 Notes on Section 1.8
8.1.8 Notes on Section 1.11
8.2 Notes on Chapter 2
8.2.1 Notes on Section 2.1
8.2.2 Notes on Section 2.2
8.2.3 Notes on Section 2.3
8.2.4 Notes on Section 2.4
8.2.5 Notes on Section 2.5
8.2.6 Notes on Section 2.7
8.2.7 Notes on Section 2.8
8.3 Notes on Chapter 3
8.3.1 Notes on Sections 3.1–3.2
8.3.2 Notes on Section 3.3
8.3.3 Notes on Section 3.4
8.3.4 Notes on Section 3.6
8.3.5 Notes on Section 3.7
8.3.6 Notes on Section 3.8
8.3.7 Notes on Section 3.9
8.3.8 Notes on Section 3.10
8.3.9 Notes on Section 3.13
8.4 Notes on Chapter 4
8.4.1 Notes on Section 4.1
8.4.2 Notes on Section 4.2
8.5 Notes on Chapter 5
8.5.1 Notes on Section 5.1
8.5.2 Notes on Section 5.2
8.5.3 Notes on Section 5.3
8.5.4 Notes on Section 5.4
8.5.5 Notes on Section 5.5
8.5.6 Notes on Section 5.6
8.5.7 Notes on Section 5.7
8.5.8 Notes on Section 5.9
8.5.9 Notes on Section 5.10
8.5.10 Notes on Section 5.12
8.5.11 Notes on Section 5.13
8.6 Notes on Chapter 6
8.6.1 Notes on Sections 6.1–6.3
8.6.2 Notes on Sections 6.4–6.5
8.6.3 Notes on Section 6.6
8.7 Notes on Chapter 7
8.7.1 Notes on Sections 7.1–7.2
8.7.2 Notes on Section 7.3
Chapter 9 Finite Variations on the Isoperimetric Problem
Chapter 10 Higher Dimensions
10.1 Existence of Economic Packings and Coverings
10.2 Upper Bounds for ?(??) and Lower Bounds for ?(??)
10.2.1 Blichfeldt’s bound
10.2.2 The simplex bound
10.2.3 The linear programming bound
10.2.4 Arrangements of points with minimum potential energy
10.2.5 Lattice arrangements of balls
10.3 Bounds for the Packing and Covering Density of Convex Bodies
10.4 The Structure of Optimal Arrangements
Chapter 11 Ball Packings in Hyperbolic Space
11.1 The Simplex Bound
11.2 Hyperspheres
11.3 Solid Arrangements
11.4 Completely Saturated Packings and Completely Reduced Coverings
11.5 A Probabilistic Approach to Optimal Arrangements and their Density
Chapter 12 Multiple Arrangements
12.1 Multiple Arrangements on the Plane
12.2 Decomposition of Multiple Arrangements
12.3 Multiple Arrangements in Space
12.4 Multiple Tiling
Chapter 13 Neighbors
13.1 The Newton Number of Convex Disks
13.2 The Hadwiger Number of Convex Disks
13.3 Translates of a Jordan Disk with a Common Point
13.4 The Number of Touching Pairs in Finite Packings
13.5 ?-Neighbor Packings
13.6 Maximal Packings
13.7 Higher-Order Neighbors
13.8 The Newton Number of Balls
13.9 ?-Neighbor Packing of Congruent Balls
13.10 Results About Convex Bodies
13.11 Mutually Touching Translates of a Convex Body
13.12 Mutually Touching Cylinders
13.13 Cylinders Touching a Ball
13.14 Neighbors in Lattice Packings
Chapter 14 Packing and Covering Properties of Sequences of Convex Bodies
14.1 Packing and Covering Cubes and Boxes
14.2 Results for General Convex Bodies
14.3 On-Line Packing and Covering
14.4 Special Convex Disks
14.5 Packing in and Covering of the Whole Space
14.6 Covering with Slabs
Chapter 15 Four Classic Problems
G. Fejes Tóth andW. Kuperberg 15.1 The Borsuk Problem
15.2 Tarski’s Plank Problem
15.3 The Kneser–Poulsen Problem
15.4 Covering a Convex Body by Smaller Homothetic Copies
Chapter 16 Miscellaneous Problems About Packing and Covering
16.1 Arranging Houses
16.2 Packing Barrels
16.3 Covering with a Margin
16.4 Finite Packing and Covering in 2 Dimensions
16.5 Finite Arrangements in Higher Dimensions
16.6 Slab, Cylinder, Torus
16.7 Close Packings and Loose Coverings
16.8 Arranging Regular Tetrahedra
16.9 Packing Cylinders
16.10 Obstructing Light
16.11 Avoiding Obstacles
16.12 Stability
16.13 Minkowskian Arrangements
16.14 Saturated Arrangements
16.15 Compact Packings
16.16 Totally Separable Packings
16.17 Point-Trapping Lattices
16.18 Connected Arrangements
16.19 Points on the Sphere
16.20 Arrangements of Great Circles
References for Part I
References for Part II
References for Chapter 8
References for Chapter 9
References for Chapter 10
References for Chapter 11
References for Chapter 12
References for Chapter 13
References for Chapter 14
References for Chapter 15
References for Chapter 16
Name Index
Subject Index
