Finite Packing and Covering.

Cover -- Half-title -- Title -- Copyright -- Dedication -- Contents -- Preface -- Notation -- Part 1 Arrangements in Dimension Two -- 1 Congruent Domains in the Euclidean Plane -- 1.1. Periodic and Finite Arrangements -- 1.2. The Hexagon Bound for Packings Inside an Octagon -- 1.3. The Hexagon Bound for Periodic Packings -- 1.4. Packings Inside Any Convex Container -- 1.5. Noncrossing Coverings -- 1.6. Packing and Covering by Similar Convex Domains -- 1.7. Coverings by Fat Ellipses -- 1.8. Minimal Perimeter and Diameter for Packings -- 1.9. Covering the Maximal Perimeter -- 1.10. Related Problems -- Separable Packings -- Snakes -- Clouds -- Newton Number -- Packing Unit Squares into Large Squares -- 2 Translative Arrangements -- 2.1. About the Minkowski Plane -- 2.2. Periodic Packing and Covering by Translates -- 2.3. Finite Packings of Centrally Symmetric Domains -- 2.4. Asymptotic Structure of Optimal Translative Packings -- 2.5. Finite and Periodic Coverings -- 2.6. The Bambah Inequality for Coverings -- 2.7. The Fejes Tóth Inequality for Coverings -- 2.8. Covering the Area by o-Symmetric Convex Domains -- 2.9. The Hadwiger Number -- 2.10. The Generalized Hadwiger Number -- 2.11. Related Problems -- Covering Any Convex Shape by Homothetic Domains -- Packing Homothetic Copies Into a Convex Domain -- Covering a Convex Domain by Homothetic Copies -- Covering the Maximal Area by Given Number of Translates -- Touching Pairs and Snakes -- Translative Clouds -- Multiple Coverings by Translates -- 3 Parametric Density -- 3.1. Planar Packings for Small… -- 3.2. Planar Packings for Reasonably Large… -- 3.3. Parametric Density for Coverings -- 3.4. Examples -- 3.4.1. Packings of Parallelograms -- 3.4.2. Finite Lattice Packings and the Wulff-Shape -- 3.4.3. Coverings by Parallelograms -- 3.4.4. Coverings by Regular Hexagons -- 3.4.5. Coverings by Ellipses.

3.4.6. Finite Lattice Coverings by Unit Discs -- 4 Packings of Circular Discs -- 4.1. Associated Cell Complexes and the Triangle Bound -- The Dirichlet-Voronoi Complex -- The Delone Complex -- Formulae and Estimates for SigmaS(r ) -- 4.2. The Absolute Version of the Thue-Groemer Inequality -- 4.3. Minimal Area for Packings of n Unit Discs -- 4.4. Packings on the Sphere -- 4.5. Packings of Spherical Discs of Radius Pi / 6 -- 4.6. Packings of Hyperbolic Circular Discs -- 4.7. Related Problems -- 5 Coverings by Circular Discs -- 5.1. The Delone Complex and the Triangle Bound -- 5.2. The Bambah Inequality and the Fejes Tóth Inequality -- 5.3. Covering the Area by Euclidean Discs -- 5.4. Coverings by Spherical Discs -- 5.5. Coverings by Hyperbolic Discs -- 5.6. The Moment Inequality and Covering the Maximal Area -- 5.7. Covering the Perimeter by Unit Discs -- 5.8. Related Problems -- Covering Some Specific Shapes in R2 -- Multiple Coverings by Unit Discs -- Coverings by Circular Discs of Different Sizes in R2 -- Solid Coverings of S2 -- Part 2 Arrangements in Higher Dimensions -- 6 Packings and Coverings by Spherical Balls -- Some Related Notions -- The Linear Programming Bound -- 6.1. Packing d + 2 Balls on Sd -- 6.2. Packing at Most 2d + 2 Balls on Sd -- 6.3. Density Bounds for Packings on Sd -- 6.4. The Simplex Bound for Packings on S3 -- 6.5. Covering Sd by at Most d + 2 Balls -- 6.6. Covering Sd by d + 3 Balls -- 6.7. Covering S3 by Eight Balls -- 6.8. Economic Coverings of Sd by Equal Balls -- 6.9. The Simplex Bound for Hyperbolic Ball Packings -- 7 Congruent Convex Bodies -- 7.1. Periodic and Finite Arrangements -- 7.2. Density for Finite Packings and Coverings -- 7.3. Minimal Mean Projection for Packings -- 7.4. Maximal Mean Width for Coverings -- 7.5. Related Problems -- Sequences of Unrelated Convex Bodies -- Newton Number.

8 Packings and Coverings by Unit Balls -- 8.1. The Newton Number -- 8.2. Periodic Arrangements of Balls -- 8.3. Finite Ball Packings of Maximal Density -- 8.4. Minimal Mean Projection for Ball Packings -- 8.5. Packings and Coverings with Respect to a Larger Ball -- 8.6. Optimal Finite Coverings by Unit Balls -- 8.7. Comments on Ball Packings and on Shapes in Nature -- 9 Translative Arrangements -- 9.1. Minkowski Geometry and the Busemann Surface Area -- 9.2. Finite Translative Tilings -- 9.3. Optimal Arrangements of a Large Number of Translates -- 9.4. Periodic Arrangements and Clusters -- 9.5. Economic Periodic Packings and Coverings by Translates -- 9.6. The Hadwiger Number -- 9.7. Lower Bound for the Hadwiger Number in Low Dimensions -- 9.8. The Generalized Hadwiger Number -- 9.9. Lower Bound for the Generalized Hadwiger Number -- 9.10. The Asymptotics of the Generalized Hadwiger Number -- 9.11. Pairwise Touching Translates and Antipodal Points -- 9.12. Economic Translational Clouds -- 9.13. Clouds Require Plentiful Balls -- 9.14. Sparse Clouds -- 9.15. The Hadwiger-Gohberg Number -- 9.16. Related Problems -- Relatives of the Hadwiger Number -- Minimizing Some Mixed Volume -- Homothetic Cubes -- Snakes of Cubes -- 10 Parametric Density -- 10.1. Sausage Packings -- 10.2. Near-Sausage Coverings -- 10.3. Cluster-like Optimal Packings and Coverings -- 10.4. Asymptotic Density for Packings and Coverings -- 10.5. The Critical Radius for Packings and Coverings -- 10.6. The Sausage Radius for Packings -- 10.7. The Critical and the Sausage Radius May Not Be Equal -- 10.8. Ball Packings -- 10.9. Packings of Parallelepipeds -- 10.10. Coverings by Parallelepipeds -- 10.11. Finite Lattice Packings and the Wulff-Shape -- Very Brief History of Geometrical Crystallography -- Finite Lattice Packings and Crystallography -- Finite Periodic Packings.

Finite Ball Packings and Quasi-periodic Sets -- 10.12. Finite Lattice Packings in Three Space -- Appendix: Background -- A.1. Some General Notions -- A.2. Convex Sets in Rd -- A.3. The Space of Compact Convex Sets in Rd -- A.4. The Spherical Space and the Hyperbolic Space -- A.5. Surfaces of Constant Curvature -- A.5.1. Convex Arcs -- A.5.2. Estimating the Area and the Perimeter -- A.5.3. Cell Complexes and the Euler Characteristic -- A.5.4. Compact Surfaces of Constant Curvature 0, 1, or -1 -- A.6. Mixed Areas in R2 -- A.7. Polyhedral Sets and Polytopes in Rd -- A.8. Associated Balls and Ellipsoids in Rd -- A.9. Volume, Surface Area, and Lebesgue Measure in Rd -- A.10. Hausdorff Measure and Lipschitz Functions in Rd -- A.11. Intrinsic Volumes in Rd -- A.12. Mixed Volumes in Rd -- A.13. Lattices in Rd and Tori -- A.14. A Little Bit of Probability -- Bibliography -- Index.