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Combinatorial Geometry

Combinatorial Geometry

János Pach, Pankaj K. Agarwal

ISBN: 978-1-118-03136-0

Oct 2011

376 pages



A complete, self-contained introduction to a powerful and resurging mathematical discipline . Combinatorial Geometry presents and explains with complete proofs some of the most important results and methods of this relatively young mathematical discipline, started by Minkowski, Fejes Toth, Rogers, and Erd???s. Nearly half the results presented in this book were discovered over the past twenty years, and most have never before appeared in any monograph. Combinatorial Geometry will be of particular interest to mathematicians, computer scientists, physicists, and materials scientists interested in computational geometry, robotics, scene analysis, and computer-aided design. It is also a superb textbook, complete with end-of-chapter problems and hints to their solutions that help students clarify their understanding and test their mastery of the material. Topics covered include:
* Geometric number theory
* Packing and covering with congruent convex disks
* Extremal graph and hypergraph theory
* Distribution of distances among finitely many points
* Epsilon-nets and Vapnik--Chervonenkis dimension
* Geometric graph theory
* Geometric discrepancy theory
* And much more

Geometry of Numbers.

Approximation of a Convex Set by Polygons.

Packing and Covering with Congruent Convex Discs.

Lattice Packing and Lattice Covering.

The Method of Cell Decomposition.

Methods of Blichfeldt and Rogers.

Efficient Random Arrangements.

Circle Packings and Planar Graphs.


Extremal Graph Theory.

Repeated Distances in Space.

Arrangement of Lines.

Applications of the Bounds on Incidences.

More on Repeated Distances.

Geometric Graphs.

Epsilon Nets and Transversals of Hypergraphs.

Geometric Discrepancy.

Hints to Exercises.