Convex PolytopesSpringer Science & Business Media, 2003 - 466 páginas "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) |
Conteúdo
Notation and prerequisites | 3 |
12 Topology | 7 |
13 Additional notes and comments | 7 |
Convex sets | 8 |
22 Support and separation | 10 |
23 Convex hulls | 14 |
24 Extreme and exposed points faces and poonems | 17 |
25 Unbounded convex sets | 23 |
99 Additional notes and comments | 170 |
Extremal Problems Concerning Numbers of Faces | 171 |
102 Lower Bounds for f i 1 in Terms of f₀ | 181 |
104 The Set fP⁴ | 189 |
105 Exercises | 195 |
106 Additional notes and comments | 197 |
Properties of boundary complexes | 198 |
111 Skeletons of simplices contained in FP | 198 |
26 Polyhedral sets | 26 |
27 Remarks | 28 |
28 Additional notes and comments | 30 |
Polytopes | 31 |
32 Combinatorial types of polytopes complexes | 38 |
33 Diagrams and Schlegel diagrams | 42 |
34 Duality of polytopes | 46 |
35 Remarks | 51 |
36 Additional notes and comments | 52 |
Examples | 53 |
42 Pyramids | 54 |
43 Bipyramids | 55 |
44 Prisms | 56 |
45 Simplicial and simple polytopes | 57 |
46 Cubical polytopes | 59 |
47 Cyclic polytopes | 61 |
48 Exercises | 63 |
49 Additional notes and comments | 69 |
Fundamental Properties and Constructions | 70 |
51 Representations of polytopes as sections or projections | 71 |
52 The inductive construction of polytopes | 78 |
53 Lower semicontinuity of the functions fkP | 83 |
54 Galetransforms and Galediagrams | 85 |
55 Existence of combinatorial types | 90 |
56 Additional notes and comments | 96 |
Polytopes with few vertices | 97 |
62 dPolytopes with d + 3 vertices | 102 |
63 Gale diagrams of polytopes with few vertices | 108 |
64 Centrally symmetric polytopes | 114 |
65 Exercises | 119 |
66 Remarks | 121 |
67 Additional notes and comments | 121 |
Neighborly polytopes | 122 |
72 dNeighborly dpolytopes | 123 |
73 Exercises | 125 |
74 Remarks | 127 |
75 Additional notes and comments | 128 |
Eulers Relation | 129 |
82 Proof of Eulers theorem | 132 |
83 A generalization of Eulers relation | 135 |
84 The Euler characteristic of complexes | 136 |
85 Exercises | 137 |
86 Remarks | 139 |
87 Additional notes and comments | 141 |
Analogues of Eulers Relation | 142 |
92 The DehnSommerville equations | 143 |
93 Quasisimplicial polytopes | 151 |
94 Cubical polytopes | 153 |
95 Solutions of the DehnSommerville equations | 158 |
96 The fvectors of neighborly dpolytopes | 160 |
97 Exercises | 166 |
98 Remarks | 168 |
112 A proof of the van KampenFlores theorem | 208 |
113 dConnectedness of the Graphs of dPolytopes | 210 |
114 Degree of total separability | 215 |
115 dDiagrams | 216 |
116 Additional notes and comments | 223 |
kEquivalence of polytopes | 224 |
122 Dimensional ambiguity | 224 |
123 Strong and weak ambiguity | 226 |
124 Additional notes and comments | 233 |
3Polytopes | 234 |
132 Consequences and analogues of Steinitzs theorem | 242 |
133 Eberhards theorem | 251 |
134 Additional results on 3realizable sequences | 269 |
135 3Polytopes with circumspheres and circumcircles | 282 |
136 Remarks | 286 |
137 Additional notes and comments | 295 |
Anglesums relations the Steiner point | 296 |
142 Anglesums relations for simplicial polytopes | 302 |
143 The Steiner point of a polytope | 305 |
144 Remarks | 310 |
145 Additional notes and comments | 314 |
Addition and decomposition of polytopes | 315 |
152 Approximation of polytopes by vector sums | 322 |
153 Blaschke addition | 329 |
154 Remarks | 335 |
155 Additional notes and comments | 339 |
Diameters of polytopes | 340 |
162 The functions and b | 345 |
163 W Paths | 352 |
164 Additional notes and comments | 354 |
Long paths and circuits on polytopes | 355 |
171 Hamiltonian paths and circuits | 355 |
172 Extremal pathlengths of polytopes | 364 |
173 Heights of polytopes | 373 |
174 Circuit codes | 379 |
175 Additional notes and comments | 388 |
Arrangements of hyperplanes | 389 |
182 2Arrangements | 395 |
183 Generalizations | 405 |
184 Additional notes and comments | 409 |
Concluding remarks | 410 |
192 kContent of polytopes | 414 |
193 Antipodality and related notions | 416 |
194 Additional notes and comments | 422 |
Tables | 423 |
Addendum | 424 |
Errata for the 1967 edition | 427 |
Bibliography | 428 |
Additional Bibliography | 447 |
Index of Terms | 473 |
465 | |