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Based on a graduate course given at the Technische Universität, Berlin, these lectures present a wealth of material on the modern theory of convex polytopes. The clear and straightforward exposition 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), and 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 will provide interesting and enjoyable reading for researchers as well as students.
Preface; Preface to the Second Printing; Introduction and Examples; 1. Polytopes, Polyhedra, and Cones; 2. Faces of Polytopes; 3. Graphs for Polytopes; 4. Steinitz' Theorem for 3-Polytopes; 5. Schlegel Diagrams for 4-Polytopes; 6. Duality, Gale Diagrams, and Applications; 7. Fans, Arrangements, Zonotopes, and Tilings; 8. Shellability and the Upper Bound Theorem; 9. Fiber Polytopes, and Beyond; References; Index.