Convex polytopes:
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| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
New York ; Berlin ; Heidelberg ; Hong Kong ; London ; Milan ; Pa
Springer
2003
|
| Edition: | 2. ed. |
| Series: | Graduate texts in mathematics
221 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Item Description: | Literaturverz. S. 429 - 448y |
| Physical Description: | XVI, 466 S. graph. Darst. : 24 cm |
| ISBN: | 0387004246 0387404090 9780387404097 |
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| 250 | |a 2. ed. |b prepared by Volker Kaibel ... | ||
| 264 | 1 | |a New York ; Berlin ; Heidelberg ; Hong Kong ; London ; Milan ; Pa |b Springer |c 2003 | |
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| 490 | 1 | |a Graduate texts in mathematics |v 221 | |
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| 650 | 4 | |a Polytopes convexes | |
| 650 | 4 | |a Convex polytopes | |
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Record in the Search Index
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| adam_text |
CONTENTS
Preface vii
Preface to the 2002 edition xi
1 Notation and prerequisites 1
1.1 Algebra 1
1.2 Topology 5
1.3 Additional notes and comments 7a
2 Convex sets 8
2.1 Definition and elementary properties 8
2.2 Support and separation 10
2.3 Convex hulls 14
2.4 Extreme and exposed points; faces and poonems 17
2.5 Unbounded convex sets 23
2.6 Polyhedral sets 26
2.7 Remarks 28
2.8 Additional notes and comments 30a
3 Polytopes 31
3.1 Definition and fundamental properties 31
3.2 Combinatorial types of polytopes; complexes 38
3.3 Diagrams and Schlegel diagrams 42
3.4 Duality of polytopes 46
3.5 Remarks 51
3.6 Additional notes and comments 52a
4 Examples 53
4.1 Thed simplex 53
4.2 Pyramids 54
4.3 Bipyramids 55
4.4 Prisms 56
4.5 Simplicial and simple polytopes 57
4.6 Cubical polytopes 59
4.7 Cyclic polytopes 61
4.8 Exercises 63
4.9 Additional notes and comments 69a
xiv CONVEX POLYTOPES
5 Fundamental properties and constructions 70
5.1 Representations of polytopes as sections or projections . 71
5.2 The inductive construction of polytopes 78
5.3 Lower semicontinuity of the functions fk(P) 83
5.4 Gale transforms and Gale diagrams 85
5.5 Existence of combinatorial types 90
5.6 Additional notes and comments 96a
6 Polytopes with few vertices 97
6.1 rf Polytopes with d+2 vertices 97
6.2 rf Polytopes with d+3 vertices 102
6.3 Gale diagrams of polytopes with few vertices 108
6.4 Centrally symmetric polytopes 114
6.5 Exercises 119
6.6 Remarks 121
6.7 Additional notes and comments 121a
7 Neighborly polytopes 122
7.1 Definition and general properties 122
7.2 [^ Neighborly d polytopes 123
7.3 Exercises 125
7.4 Remarks 127
7.5 Additional notes and comments 129a
8 Euler's relation 130
8.1 Euler's theorem 130
8.2 Proof of Euler's theorem 134
8.3 A generalization of Euler's relation 137
8.4 The Euler characteristic of complexes 138
8.5 Exercises 139
8.6 Remarks 141
8.7 Additional notes and comments 142a
9 Analogues of Euler's relation 143
9.1 The incidence equation 143
9.2 The Dehn Sommerville equations 145
9.3 Quasi simplicial polytopes 153
9.4 Cubical polytopes . 155
9.5 Solutions of the Dehn Sommerville equations 160
9.6 The / vectors of neighborly d polytopes 162
9.7 Exercises 168
CONTENTS XV
9.8 Remarks 170
9.9 Additional notes and comments 171a
10 Extremal problems concerning numbers of faces 172
10.1 Upper bounds for /), i 1, in terms of /„ 172
10.2 Lower bounds for/), i 1, in terms of/„ 183
10.3 The sets/(^»3) and/(^3) 189
10.4 The set /(£»4) 191
10.5 Exercises 197
10.6 Additional notes and comments 198a
11 Properties of boundary complexes 199
11.1 Skeletons of simplices contained in 39{P) 200
11.2 A proof of the van Kampen Flores theorem 210
11.3 (/ Connectedness of the graphs of rf polytopes 212
11.4 Degree of total separability 217
11.5 (/ Diagrams 218
11.6 Additional notes and comments 224a
12 ft Equivalenceofpolytopes 225
12.1 it Equivalence and ambiguity 225
12.2 Dimensional ambiguity 226
12.3 Strong and weak ambiguity 228
12.4 Additional notes and comments 234a
13 3 Polytopes 235
13.1 Steinitz's theorem 235
13.2 Consequences and analogues of Steinitz's theorem 244
13.3 Eberhard's theorem 253
13.4 Additional results on 3 realizable sequences 271
13.5 3 Polytopes with circumspheres and circumcircles 284
13.6 Remarks 288
13.7 Additional notes and comments 296a
14 Angle sums relations; the Steiner point 297
14.1 Gram's relation for angle sums 297
14.2 Angle sums relations for simplicial polytopes 304
14.3 The Steiner point of a polytope (by G. C. Shephard) 307
14.4 Remarks 312
14.5 Additional notes and comments 315a
xvi CONVEX POLYTOPES
15 Addition and decomposition of poly topes (by G. C. Shephard) . 316
15.1 Vector addition 316
15.2 Approximation of polytopes by vector sums 324
15.3 Blaschke addition 331
15.4 Remarks 337
15.5 Additional notes and comments 340a
16 Diameters of polytopes (by Victor Klee) 341
16.1 Extremal diameters of / polytopes 342
16.2 The functions A and Ab 347
16.3 Wv Paths 354
16.4 Additional notes and comments 355a
17 Long paths and circuits on polytopes (by Victor Klee) 356
17.1 Hamiltonian paths and circuits 357
17.2 Extremal path lengths of polytopes 366
17.3 Heights of polytopes 375
17.4 Circuit codes 381
17.5 Additional notes and comments 389a
18 Arrangements of hyperplanes 390
18.1 d Arrangements 390
18.2 2 Arrangements 397
18.3 Generalizations 407
18.4 Additional notes and comments 410a
19 Concluding remarks 411
19.1 Regular polytopes and related notions 411
19.2 Jt Content of polytopes 416
19.3 Antipodality and related notions 418
19.4 Additional notes and comments 423a
Tables 424
Addendum 426
Errata forthe 1967 edition 428a
Bibliography 429
Additional Bibliography . 448a
Index of Terms 449
Index of Symbols 467 |
| any_adam_object | 1 |
| author | Grünbaum, Branko 1929-2018 |
| author_GND | (DE-588)1067945075 (DE-588)118199935 |
| author_facet | Grünbaum, Branko 1929-2018 |
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| dewey-full | 516.3/5 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 516 - Geometry |
| dewey-raw | 516.3/5 |
| dewey-search | 516.3/5 |
| dewey-sort | 3516.3 15 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| edition | 2. ed. |
| format | Book |
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| indexdate | 2025-04-02T08:00:41Z |
| institution | BVB |
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| language | English |
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| series2 | Graduate texts in mathematics |
| spelling | Grünbaum, Branko 1929-2018 Verfasser (DE-588)1067945075 aut Convex polytopes Branko Grünbaum 2. ed. prepared by Volker Kaibel ... New York ; Berlin ; Heidelberg ; Hong Kong ; London ; Milan ; Pa Springer 2003 XVI, 466 S. graph. Darst. : 24 cm txt rdacontent n rdamedia nc rdacarrier Graduate texts in mathematics 221 Literaturverz. S. 429 - 448y Geometria elementar larpcal Poliedros (combinatória) larpcal Poliedros (geometria) larpcal Polytopes convexes Convex polytopes Konvexität (DE-588)4114284-6 gnd rswk-swf Polytop (DE-588)4175324-0 gnd rswk-swf Konvexes Polytop (DE-588)4367579-7 gnd rswk-swf Konvexes Polytop (DE-588)4367579-7 s DE-604 Konvexität (DE-588)4114284-6 s 1\p DE-604 Polytop (DE-588)4175324-0 s 2\p DE-604 Kaibel, Volker 1969- Sonstige (DE-588)118199935 oth Graduate texts in mathematics 221 (DE-604)BV000000067 221 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=010250164&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk 2\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Grünbaum, Branko 1929-2018 Convex polytopes Graduate texts in mathematics Geometria elementar larpcal Poliedros (combinatória) larpcal Poliedros (geometria) larpcal Polytopes convexes Convex polytopes Konvexität (DE-588)4114284-6 gnd Polytop (DE-588)4175324-0 gnd Konvexes Polytop (DE-588)4367579-7 gnd |
| subject_GND | (DE-588)4114284-6 (DE-588)4175324-0 (DE-588)4367579-7 |
| title | Convex polytopes |
| title_auth | Convex polytopes |
| title_exact_search | Convex polytopes |
| title_full | Convex polytopes Branko Grünbaum |
| title_fullStr | Convex polytopes Branko Grünbaum |
| title_full_unstemmed | Convex polytopes Branko Grünbaum |
| title_short | Convex polytopes |
| title_sort | convex polytopes |
| topic | Geometria elementar larpcal Poliedros (combinatória) larpcal Poliedros (geometria) larpcal Polytopes convexes Convex polytopes Konvexität (DE-588)4114284-6 gnd Polytop (DE-588)4175324-0 gnd Konvexes Polytop (DE-588)4367579-7 gnd |
| topic_facet | Geometria elementar Poliedros (combinatória) Poliedros (geometria) Polytopes convexes Convex polytopes Konvexität Polytop Konvexes Polytop |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=010250164&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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