Lagerungen: Arrangements in the Plane, on the Sphere, and in Space

Description

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. 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 𝒏 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 𝒏 = 7 6.3 The Maximal Configuration for 𝒏 = 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 𝜹(𝑩𝒏) and Lower Bounds for 𝝑(𝑩𝒏) 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 𝒏-Neighbor Packings 13.6 Maximal Packings 13.7 Higher-Order Neighbors 13.8 The Newton Number of Balls 13.9 𝒏-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