Combinatorial Geometry: 37

How many objects of a given shape and size can be packed into a large box of fixed volume? Can one plant n trees in an orchard, not all along the same line, so that every line determined by two trees will pass through a third? These questions, raised by Hilbert and Sylvester roughly one hundred years ago, have generated a lot of interest among professional and amateur mathematicians and scientists. They have led to the birth of a new mathematical discipline with close ties to classical geometry and number theory, and with many applications in coding theory, potential theory, computational geometry, computer graphics, robotics, etc. Combinatorial Geometry offers a self-contained introduction to this rapidly developing field, where combinatorial and probabilistic (counting) methods play a crucial role. This book has grown out of the material of both undergraduate and graduate courses in mathematics and computer science given by János Pach at the Courant Institute of Mathematical Sciences, New York University. Divided into two parts― Arrangements of Convex Sets and Arrangements of Points and Lines―it presents and explains some of the most important and ingenious results in combinatorial geometry, including:

• Dowker’s theorems
• Fáry’s theorem
• Fejes Tóth’ stheorems
• Methods of Blichtfeldt and Rogers
• Minkowski―Hlawka theorem
• Koebe’s representation theorem
• Lipton―Tarjan separator theorem
• Theorems of Erd’s, Turán, and Ramsey
• Szemerédi’s regularity lemma
• Szemerédi―Trotter theorems
• Methods of Clarkson, Edelsbrunner, Guibas, Sharir, and Welzl
• Counterexample to Borsuk’s conjecture

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 Tóth, 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