It was written by Branko Grünbaum, with contributions from Victor Klee, Micha Perles, and G. C. Shephard, and published in 1967 by John Wiley & Sons.
[5][6] A second edition, prepared with the assistance of Volker Kaibel, Victor Klee, and Günter M. Ziegler, was published by Springer-Verlag in 2003, as volume 221 of their book series Graduate Texts in Mathematics.
After two chapters introducing background material in linear algebra, topology, and convex geometry, two more chapters provide basic definitions of polyhedra, in their two dual versions (intersections of half-spaces and convex hulls of finite point sets), introduce Schlegel diagrams, and provide some basic examples including the cyclic polytopes.
[7][8] 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
[1] 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".
[2] 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.