Convex Polytopes

Capa
Springer 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
Index of Symbols
465
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