This volume studies convex sets, polytopes, and discrete geometric structures.
This volume studies convex sets, polytopes, and discrete geometric structures. It emphasizes combinatorial structure, geometric inequalities, and computational aspects.
Part I. Convex Sets
Chapter 1. Convexity
1.1 Convex sets and combinations 1.2 Convex hulls 1.3 Extreme points 1.4 Carathéodory theorem 1.5 Examples
Chapter 2. Separation and Support
2.1 Supporting hyperplanes 2.2 Separation theorems 2.3 Dual cones 2.4 Applications 2.5 Examples
Chapter 3. Convex Functions
3.1 Definitions 3.2 Properties 3.3 Jensen inequality 3.4 Subgradients 3.5 Applications
Part II. Polytopes
Chapter 4. Convex Polytopes
4.1 Definitions 4.2 Vertices, edges, faces 4.3 Face lattices 4.4 Examples 4.5 Applications
Chapter 5. Polyhedral Theory
5.1 Half-space representations 5.2 Minkowski theorem 5.3 Dual polytopes 5.4 Examples 5.5 Applications
Chapter 6. Combinatorics of Polytopes
6.1 f-vectors 6.2 Euler relations 6.3 Shellability 6.4 Applications 6.5 Examples
Part III. Discrete Geometry
Chapter 7. Lattices
7.1 Lattice points 7.2 Basis and determinants 7.3 Geometry of numbers 7.4 Applications 7.5 Examples
Chapter 8. Packing and Covering
8.1 Sphere packing 8.2 Covering problems 8.3 Density bounds 8.4 Applications 8.5 Examples
Chapter 9. Combinatorial Geometry
9.1 Incidence problems 9.2 Arrangements 9.3 Extremal problems 9.4 Applications 9.5 Examples
Part IV. Geometric Inequalities
Chapter 10. Classical Inequalities
10.1 Isoperimetric inequality 10.2 Brunn–Minkowski inequality 10.3 Applications 10.4 Examples 10.5 Consequences
Chapter 11. Convex Geometry Inequalities
11.1 Minkowski inequalities 11.2 Mixed volumes 11.3 Applications 11.4 Examples 11.5 Connections
Chapter 12. Functional Inequalities
12.1 Log-concavity 12.2 Concentration inequalities 12.3 Applications 12.4 Examples 12.5 Connections
Part V. Algorithms and Computation
Chapter 13. Convex Optimization Geometry
13.1 Geometry of feasible sets 13.2 Polyhedral methods 13.3 Applications 13.4 Examples 13.5 Connections
Chapter 14. Computational Geometry
14.1 Convex hull algorithms 14.2 Voronoi diagrams 14.3 Delaunay triangulations 14.4 Applications 14.5 Examples
Chapter 15. Integer Geometry
15.1 Integer points in polytopes 15.2 Ehrhart theory (overview) 15.3 Applications 15.4 Examples 15.5 Connections
Part VI. Advanced Topics
Chapter 16. Random Polytopes
16.1 Definitions 16.2 Probabilistic methods 16.3 Applications 16.4 Examples 16.5 Connections
Chapter 17. Discrete Differential Geometry (Overview)
17.1 Discrete curvature 17.2 Graph embeddings 17.3 Applications 17.4 Examples 17.5 Connections
Chapter 18. Geometric Measure Links
18.1 Measures on convex sets 18.2 Volume theory 18.3 Applications 18.4 Examples 18.5 Connections
Part VII. Applications
Chapter 19. Optimization and Operations Research
19.1 Linear programming geometry 19.2 Feasible regions 19.3 Applications 19.4 Examples 19.5 Connections
Chapter 20. Computer Science
20.1 Data structures 20.2 Algorithmic geometry 20.3 Complexity 20.4 Applications 20.5 Examples
Chapter 21. Physics and Materials
21.1 Crystallography 21.2 Packing in materials 21.3 Energy minimization 21.4 Applications 21.5 Examples
Part VIII. Research Directions
Chapter 22. Advanced Topics
22.1 High-dimensional convexity 22.2 Asymptotic geometry 22.3 Random structures 22.4 Modern developments 22.5 Emerging areas
Chapter 23. Open Problems
23.1 Packing bounds 23.2 Polytope classification 23.3 Incidence geometry limits 23.4 Computational challenges 23.5 Future directions
Chapter 24. Historical and Conceptual Notes
24.1 Development of convex geometry 24.2 Key contributors 24.3 Evolution of discrete geometry 24.4 Cross-disciplinary impact 24.5 Summary
Appendix
A. Convex geometry formulas B. Polytope reference tables C. Proof techniques checklist D. Algorithm templates E. Cross-reference to other MSC branches