Convex Polytopes

Capa
Springer Science & Business Media, 2003 - 466 páginas
1 Resenha

"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)

 

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Conteúdo

Notation and prerequisites
1
12 Topology
5
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
Extremal Problems Concerning Numbers of Faces
172
102 Lower Bounds for f i 1 in Terms of f₀
183
104 The Set fP⁴
191
105 Exercises
197
106 Additional notes and comments
198
Properties of boundary complexes
199
111 Skeletons of simplices contained in FP
200
112 A proof of the van KampenFlores theorem
210

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
129
Eulers Relation
130
82 Proof of Eulers theorem
134
83 A generalization of Eulers relation
137
84 The Euler characteristic of complexes
138
85 Exercises
139
86 Remarks
141
87 Additional notes and comments
142
Analogues of Eulers Relation
143
92 The DehnSommerville equations
145
93 Quasisimplicial polytopes
153
94 Cubical polytopes
155
95 Solutions of the DehnSommerville equations
160
96 The fvectors of neighborly dpolytopes
162
97 Exercises
168
98 Remarks
170
99 Additional notes and comments
171
113 dConnectedness of the Graphs of dPolytopes
212
114 Degree of total separability
217
115 dDiagrams
218
116 Additional notes and comments
224
kEquivalence of polytopes
225
122 Dimensional ambiguity
226
123 Strong and weak ambiguity
228
124 Additional notes and comments
234
3Polytopes
235
132 Consequences and analogues of Steinitzs theorem
244
133 Eberhards theorem
253
134 Additional results on 3realizable sequences
271
135 3Polytopes with circumspheres and circumcircles
284
136 Remarks
288
137 Additional notes and comments
296
Anglesums relations the Steiner point
297
142 Anglesums relations for simplicial polytopes
304
143 The Steiner point of a polytope
307
144 Remarks
312
145 Additional notes and comments
315
Addition and decomposition of polytopes
316
152 Approximation of polytopes by vector sums
324
153 Blaschke addition
331
154 Remarks
337
155 Additional notes and comments
340
Diameters of polytopes
341
161 Extremal diameters of dpolytopes
342
162 The functions and b
347
163 W Paths
354
164 Additional notes and comments
355
Long paths and circuits on polytopes
356
171 Hamiltonian paths and circuits
357
172 Extremal pathlengths of polytopes
366
173 Heights of polytopes
375
174 Circuit codes
381
175 Additional notes and comments
389
Arrangements of hyperplanes
390
182 2Arrangements
397
183 Generalizations
407
184 Additional notes and comments
410
Concluding remarks
411
192 kContent of polytopes
416
193 Antipodality and related notions
418
194 Additional notes and comments
423
Tables
424
Addendum
426
Errata for the 1967 edition
428
Bibliography
429
Additional Bibliography
448
Index of Terms
449
Index of Symbols
467
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Stochastische Geometrie
Rolf Schneider,Wolfgang Weil
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