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This book is the result of a 25-year-old project and comprises a collection of more than 500 attractive open problems in the field. The largely self-contained chapters provide a broad overview of discrete geometry, along with historical details and the most important partial results related to these problems. This book is intended as a source book for both professional mathematicians and graduate students who love beautiful mathematical questions, are willing to spend sleepless nights thinking about them, and who would like to get involved in mathematical research.

0. Definitions and Notations .
1. Density Problems for Packings and Coverings
1.1 Basic Questions and Definitions
1.2 The Least Economical Convex Sets for Packing
1.3 The Least Economical Convex Sets for Covering
1.4 How Economical Are the Lattice Arrangements?
1.5 Packing with Semidisks, and the Role of Symmetry
1.6 Packing Equal Circles into Squares, Circles, Spheres
1.7 Packing Equal Circles or Squares in a Strip
1.8 The Densest Packing of Spheres
1.9 The Densest Packings of Specific Convex Bodies
1.10 Linking Packing and Covering Densities
1.11 Sausage Problems and Catastrophes
2. Structural Packing and Covering Problems
2.1 Decomposition of Multiple Packings and Coverings
2.2 Solid and Saturated Packings and Reduced Coverings
2.3 Stable Packings and Coverings
2.4 Kissing and Neighborly Convex Bodies
2.5 Thin Packings with Many Neighbors
2.6 Permeability and Blocking Light Rays
3. Packing and Covering with Homothetic Copies
3.1 Potato Bag Problems
3.2 Covering a Convex Body with Its Homothetic Copies
3.3 Levi-Hadwiger Covering Problem and Illumination
3.4 Covering a Ball by Slabs
3.5 Point Trapping and Impassable Lattice Arrangements
4. Tiling Problems
4.1 Tiling the Plane with Congruent Regions
4.2 Aperiodic Tilings and Tilings with Fivefold Symmetry
4.3 Tiling Space with Polytopes
5. Distance Problems
5.1 The Maximum Number of Unit Distances in the Plane
5.2 The Number of Equal Distances in Other Spaces
5.3 The Minimum Number of Distinct Distances in the Plane
5.4 The Number of Distinct Distances in Other Spaces
5.5 Repeated Distances in Point Sets in General Position
5.6 Repeated Distances in Point Sets in Convex Position
5.7 Frequent Small Distances and Touching Pairs
5.8 Frequent Large Distances
5.9 Chromatic Number of Unit-Distance Graphs
5.10 Further Problems on Repeated Distances ..
5.11 Integral or Rational Distances
6. Problems on Repeated Subconfigurations
6.1 Repeated Simplices and Other Patterns
6.2 Repeated Directions, Angles, Areas
6.3 Euclidean Ramsey Problems
7. Incidence and Arrangement Problems
7.1 The Maximum Number of Incidences
7.2 Sylvester-Gallai-Type Problems
7.3 Line Arrangements Spanned by a Point Set
8. Problems on Points in General Position
8.1 Structure of the Space of Order Types
8.2 Convex Polygons and the Erdos-Szekeres Problem
8.3 Halving Lines and Related Problems
8.4 Extremal Number of Special Subconfigurations
8.5 Other Problems on Points in General Position
9. Graph Drawings and Geometric Graphs
9.1 Graph Drawings
9.2 Drawing Planar Graphs
9.3 The Crossing Number
9.4 Other Crossing Numbers
9.5 From Thrackles to Forbidden Geometric Subgraphs
9.6 Further Turan-Type Problems
9.7 Ramsey-Type Problems
9.8 Geometric Hypergraphs
10. Lattice Point Problems
10.1 Packing'La. ice Points in Suhspaces
10.2 Covering Lattice Points by Subspaces
10.3 Sets of Lattice Points Avoiding Other Regularities
10.4 Visibility Problems for Lattice Points
11. Geometric Inequalities
11.1 Isoperimetric Inequalities for Polygons and Polytopes
11.2 Heilbronn-Type Problems
11.3 Circumscribed and Inscribed Convex Sets
11.4 Universal Covers
11.5 Approximation Problems
12. Index
12.1 Author Index
12.2 Subject Index
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