Based on a graduate course given at the Technische Universitat, Berlin, these lectures present a wealth of material on the modern theory of convex polytopes. The clear and straightforward presentation features many illustrations, and provides complete proofs for most theorems. The material requires only linear algebra as a prerequisite, but takes the reader quickly from the basics to topics of recent research, including a number of unanswered questions. The lectures - introduce the basic facts about polytopes, with an emphasis on the methods that yield the results (Fourier-Motzkin elimination, Schlegel diagrams, shellability, Gale transforms, and oriented matroids) - discuss important examples and elegant constructions (cyclic and neighborly polytopes, zonotopes, Minkowski sums, permutahedra and associhedra, fiber polytopes, and the Lawrence construction) - show the excitement of current work in the field (Kalai's new diameter bounds, construction of non-rational polytopes, the Bohne-Dress tiling theorem, the upper-bound theorem), and nonextendable shellings) They should provide interesting and enjoyable reading for researchers as well as students.
Based on a graduate course given at the Technische Universitat, Berlin, these lectures present a wealth of material on the modern theory of convex polytopes. The clear and straightforward presentation features many illustrations, and provides complete proofs for most theorems. The material requires only linear algebra as a prerequisite, but takes the reader quickly from the basics to topics of recent research, including a number of unanswered questions. The lectures - introduce the basic facts about polytopes, with an emphasis on the methods that yield the results (Fourier-Motzkin elimination, Schlegel diagrams, shellability, Gale transforms, and oriented matroids) - discuss important examples and elegant constructions (cyclic and neighborly polytopes, zonotopes, Minkowski sums, permutahedra and associhedra, fiber polytopes, and the Lawrence construction) - show the excitement of current work in the field (Kalai's new diameter bounds, construction of non-rational polytopes, the Bohne-Dress tiling theorem, the upper-bound theorem), and nonextendable shellings) They should provide interesting and enjoyable reading for researchers as well as students.
and Examples.- Polytopes, Polyhedra, and Cones.- Faces of Polytopes.- Graphs of Polytopes.- Steinitz’ Theorem for 3-Polytopes.- Schlegel Diagrams for 4-Polytopes.- Duality, Gale Diagrams, and Applications.- Fans, Arrangements, Zonotopes, and Tilings.- Shellability and the Upper Bound Theorem.- Fiber Polytopes, and Beyond.- Fiber Polytopes, and Beyond.
Springer Book Archives
From the reviews: "This is an excellent book on convex polytopes written by a young and extremely active researcher." (Acta Scientiarum Mathematicarum) "From the publication of the first printing, in 1994, this book became one of the most widely used textbooks in Discrete Geometry. The reviewer sees at least two reasons for that: the beautiful mathematics presented here, and the fact that the book can be used at a wide variety of levels, for several different courses. ! It is not only students who can benefits from the book. Researchers will find its updates notes and references very helpful." (Miklos Bona, MathDL, August, 2007)
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