Convex Polytopes

• 1 Notation and prerequisites.- 1.1 Algebra.- 1.2 Topology.- 1.3 Additional notes and comments.- 2 Convex sets.- 2.1 Definition and elementary properties.- 2.2 Support and separation.- 2.3 Convex hulls.- 2.4 Extreme and exposed points; faces and poonems.- 2.5 Unbounded convex sets.- 2.6 Polyhedral sets.- 2.7 Remarks.- 2.8 Additional notes and comments.- 3 Polytopes.- 3.1 Definition and fundamental properties.- 3.2 Combinatorial types of polytopes; complexes.- 3.3 Diagrams and Schlegel diagrams.- 3.4 Duality of polytopes.- 3.5 Remarks.- 3.6 Additional notes and comments.- 4 Examples.- 4.1 The d-simplex.- 4.2 Pyramids.- 4.3 Bipyramids.- 4.4 Prisms.- 4.5 Simplicial and simple polytopes.- 4.6 Cubical polytopes.- 4.7 Cyclic polytopes.- 4.8 Exercises.- 4.9 Additional notes and comments.- 5 Fundamental properties and constructions.- 5.1 Representations of polytopes as sections or projections.- 5.2 The inductive construction of polytopes.- 5.3 Lower semicontinuity of the functions fk(P).- 5.4 Gale-transforms and Gale-diagrams.- 5.5 Existence of combinatorial types.- 5.6 Additional notes and comments.- 6 Polytopes with few vertices.- 6.1 d-Polytopes with d + 2 vertices.- 6.2 d-Polytopes with d + 3 vertices.- 6.3 Gale diagrams of polytopes with few vertices.- 6.4 Centrally symmetric polytopes.- 6.5 Exercises.- 6.6 Remarks.- 6.7 Additional notes and comments.- 7 Neighborly polytopes.- 7.1 Definition and general properties.- 7.2 % MathType!MTEF!2!1!+-% feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeaaciGaaiaabeqaamaabaabaaGcbaWaamWaaeaadG% aGmUaaaeacaYOaiaiJigdaaeacaYOaiaiJikdaaaacbiGaiaiJ-rga% aiaawUfacaGLDbaaaaa!40CC!$$\left[ {\frac{1}{2}d} \right]$$-Neighborly d-polytopes.- 7.3 Exercises.- 7.4 Remarks.- 7.5 Additional notes and comments.- 8 Euler’s relation.- 8.1 Euler’s theorem.- 8.2 Proof of Euler’s theorem.- 8.3 A generalization of Euler’s relation.- 8.4 The Euler characteristic of complexes.- 8.5 Exercises.- 8.6 Remarks.- 8.7 Additional notes and comments.- 9 Analogues of Euler’s relation.- 9.1 The incidence equation.- 9.2 The Dehn-Sommerville equations.- 9.3 Quasi-simplicial polytopes.- 9.4 Cubical polytopes.- 9.5 Solutions of the Dehn-Sommerville equations.- 9.6 The f-vectors of neighborly d-polytopes.- 9.7 Exercises.- 9.8 Remarks.- 9.9 Additional notes and comments.- 10 Extremal problems concerning numbers of faces.- 10.1 Upper bounds for fi, i ? 1, in terms of fo.- 10.2 Lower bounds for fi, i ? 1, in terms of fo.- 10.3 The sets f(P3) and f(PS3).- 10.4 The set fP4).- 10.5 Exercises.- 10.6 Additional notes and comments.- 11 Properties of boundary complexes.- 11.1 Skeletons of simplices contained in ?(P).- 11.2 A proof of the van Kampen-Flores theorem.- 11.3 d-Connectedness of the graphs of d-polytopes.- 11.4 Degree of total separability.- 11.5 d-Diagrams.- 11.6 Additional notes and comments.- 12 k-Equivalence of polytopes.- 12.1 k-Equivalence and ambiguity.- 12.2 Dimensional ambiguity.- 12.3 Strong and weak ambiguity.- 12.4 Additional notes and comments.- 13 3-Polytopes.- 13.1 Steinitz’s theorem.- 13.2 Consequences and analogues of Steinitz’s theorem.- 13.3 Eberhard’s theorem.- 13.4 Additional results on 3-realizable sequences.- 13.5 3-Polytopes with circumspheres and circumcircles.- 13.6 Remarks.- 13.7 Additional notes and comments.- 14 Angle-sums relations; the Steiner point.- 14.1 Gram’s relation for angle-sums.-14.2 Angle-sums relations for simplicial polytopes.- 14.3 The Steiner point of a polytope (by G. C. Shephard).- 14.4 Remarks.- 14.5 Additional notes and comments.- 15 Addition and decomposition of polytopes.- 15.1 Vector addition.- 15.2 Approximation of polytopes by vector sums.- 15.3 Blaschke addition.- 15.4 Remarks.- 15.5 Additional notes and comments.- 16 Diameters of polytopes (by Victor Klee).- 16.1 Extremal diameters of d-polytopes.- 16.2 The functions ? and ?b.- 16.3 Wv Paths.- 16.4 Additional notes and comments.- 17 Long paths and circuits on polytopes.- 17.1 Hamiltonian paths and circuits.- 17.2 Extremal path-lengths of polytopes.- 17.3 Heights of polytopes.- 17.4 Circuit codes.- 17.5 Additional notes and comments.- 18 Arrangements of hyperplanes.- 18.1 d-Arrangements.- 18.2 2-Arrangements.- 18.3 Generalizations.- 18.4 Additional notes and comments.- 19 Concluding remarks.- 19.1 Regular polytopes and related notions.- 19.2 k-Content of polytopes.- 19.3 Antipodality and related notions.- 19.4 Additional notes and comments.- Tables.- Addendum.- Errata for the 1967 edition.- Additional Bibliography.- Index of Terms.- Index of Symbols.

"The appearance of Grunbaum'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 Grunbaum 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) From the reviews of the second edition: "Branko Grunbaum's book is a classical monograph on convex polytopes ... . As was noted by many researchers, for many years the book provided a central reference for work in the field and inspired a whole generation of specialists in polytope theory. ... Every chapter of the book is supplied with a section entitled 'Additional notes and comments' ... these notes summarize the most important developments with respect to the topics treated by Grunbaum. ... The new edition ... is an excellent gift for all geometry lovers." (Alexander Zvonkin, Mathematical Reviews, 2004b)