be combined to produce other polytopes. Chapters 16 and 17 study shortest paths and the Hirsch conjecture, longest paths and Hamiltonian cycles, and the shortness exponent of polytopes. Chapter 18 studies arrangements of hyperplanes and their dual relation to the combinatorial structure of zonotopes. A concluding chapter, chapter 19, also includes material on the symmetries of polytopes. Exercises throughout the book make it usable as a textbook, and provide additional links to recent research, and the later chapters of the book also list many open research problems. The second edition of the book keeps the content, organization, and pagination of the first edition intact, adding to it notes at the ends of each chapter on updates to the material in that chapter. These updates include material on Mnëv's universality theorem and its relation to the realizability of polytopes from their combinatorial structures, the proof of the -conjecture for simplicial spheres, and Kalai's conjecture. The second edition also provides an improved bibliography. Topics that are important to the theory of convex polytopes but not well-covered in the book Convex Polytopes include Hilbert's third problem and the theory of Dehn invariants. Audience and reception Although written at a graduate level, the main prerequisites for reading the book are linear algebra and general topology, both at an undergraduate level. In a review of the first edition of the book, Werner Fenchel calls it "a remarkable achievement", "a wealth of material", "well organized and presented in a lucid style". Over 35 years later, in giving the Steele Prize to Grünbaum for Convex Polytopes, the American Mathematical Society wrote that the book "has served both as a standard reference and as an inspiration", that it was in large part responsible for the vibrant ongoing research in polyhedral combinatorics, and that it remained relevant to this area. Reviewing and welcoming the second edition, Peter McMullen wrote that despite being "immediately rendered obsolete" by the research that it sparked, the book was still essential reading for researchers in this area. See also List of books about polyhedra References Polytopes Convex geometry Graduate Texts in Mathematics 1967 non-fiction books 2003 non-fiction books