Convex Polytopes: Second Edition Prepared by Volker Kaibel, Victor Klee, and Günter Ziegler (Graduate Texts in Mathematics)

"The appearance of Grünbaum's book **Convex Polytopes** in 1967 was a moment of grace to geometers and combinatorialists. The special spirit of the book is very much alive even in those chapters where the book's immense influence made them quickly obsolete. Some other chapters promise beautiful unexplored land for future research. The appearance of the new edition is going to be another moment of grace. Kaibel, Klee and Ziegler were able to update the convex polytope saga in a clear, accurate, lively, and inspired way." __(Gil Kalai, The Hebrew University of Jerusalem)__ "The original book of Grünbaum has provided the central reference for work in this active area of mathematics for the past 35 years...I first consulted this book as a graduate student in 1967; yet, even today, I am surprised again and again by what I find there. It is an amazingly complete reference for work on this subject up to that time and continues to be a major influence on research to this day." __(Louis J. Billera, Cornell University)__ "The original edition of **Convex Polytopes** inspired a whole generation of grateful workers in polytope theory. Without it, it is doubtful whether many of the subsequent advances in the subject would have been made. The many seeds it sowed have since grown into healthy trees, with vigorous branches and luxuriant foliage. It is good to see it in print once again." __(Peter McMullen, University College London)__ "The appearance of Grünbaum's book Convex Polytopes in 1967 was a moment of grace to geometers and combinatorialists. The special spirit of the book is very much alive even in those chapters where the book's immense influence made them quickly obsolete. Some other chapters promise beautiful unexplored land for future research. The appearance of the new edition is going to be another moment of grace. Kaibel, Klee and Ziegler were able to update the convex polytope saga in a clear, accurate, lively, and inspired way." (Gil Kalai, The Hebrew University of Jerusalem) "The original book of Grünbaum has provided the central reference for work in this active area of mathematics for the past 35 years ... I first consulted this book as a graduate student in 1967; yet, even today, I am surprised again and again by what I find there. It is an amazingly complete reference for work on this subject up to that time and continues to be a major influence on research to this day." (Louis J. Billera, Cornell University) "The original edition of Convex Polytopes inspired a whole generation of grateful workers in polytope theory. Without it, it is doubtful whether many of the subsequent advances in the subject would have been made. The many seeds it sowed have since grown into healthy trees, with vigorous branches and luxuriant foliage. It is good to see it in print once again." (Peter McMullen, University College London) Front Matter....Pages i-xvi Notation and Prerequisites....Pages 1-9 Convex Sets....Pages 10-34 Polytopes....Pages 35-60 Examples....Pages 61-79 Fundamental Properties and Constructions....Pages 80-108 Polytopes with Few Vertices....Pages 109-135 Neighborly Polytopes....Pages 136-145 Euler’s Relation....Pages 146-160 Analogues of Euler’s Relation....Pages 161-191 Extremal Problems Concerning Numbers of Faces....Pages 192-222 Properties of Boundary Complexes....Pages 223-250 k -Equivalence of Polytopes....Pages 251-262 3-Polytopes....Pages 263-328 Angle-sums Relations; the Steiner Point....Pages 329-349 Addition and Decomposition of Polytopes....Pages 350-377 Diameters of Polytopes....Pages 379-395 Long Paths and Circuits on Polytopes....Pages 396-431 Arrangements of Hyperplanes....Pages 432-454 Concluding Remarks....Pages 455-489 Back Matter....Pages 474-547 Notation And Prerequisites -- Convex Sets -- Polytopes -- Examples -- Fundamental Properties And Constructions -- Polytopes With Few Vertices -- Neighborly Polytopes -- Euler's Relation -- Analogues Of Euler's Relation -- Analogues Of Euler's Relation -- Extremal Problems Concerning Numbers Of Faces -- Properties Of Boundary Complexes -- K-equivalence Of Polytopes -- 3-polytopes -- Angle-sums Relations; The Steiner Point -- Addition And Decomposition Of Polytopes (by G.c. Shephard) -- Diameters Of Polytopes (by Victor Klee) -- Long Paths And Circuits And Polytopes (by Victor Klee) -- Arrangements Of Hyperplanes -- Concluding Remarks. Branko Grünbaum. Includes Bibliographical References (p. 429-448) And Index.

"The original edition [...] inspired a whole generation of grateful workers in polytope theory. Without it, it is doubtful whether many of the subsequent advances in the subject would have been made. The many seeds it sowed have since grown into healthy trees, with vigorous branches and luxuriant foliage. It is good to see it in print once again." —Peter McMullen, University College London

Although parts of Grünbaum's seminal work on convex polytopes were quickly outdated after its original publication in 1967, by virtue of its influence on a generation of researchers, much remains of great interest to mathematicians With few exceptions, we shall be concerned with convexity in Rd, the d-dimensional real Euclidean space.