Realization spaces of polytopes:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | German |
| Published: |
Berlin [u.a.]
Springer
1996
|
| Series: | Lecture notes in mathematics
1643 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Item Description: | Teilw. zugl.: Berlin, Techn. Univ., Habil-Schr., 1995 |
| Physical Description: | XI, 187 S. graph. Darst. |
| ISBN: | 3540620842 |
Staff View
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Record in the Search Index
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| adam_text | Contents
Preface vii
Introduction 1
1 Polytopes and their Realizations 1
1.1 Polytopes 1
1.2 History I: Steinitz s Theorem 3
1.3 History II: Polytopes in Dimension Higher than 3 4
1.4 New Results on 4 Polytopes 6
1.5 Polytopal Tools 7
1.6 Sketch of the Proof of the Universality Theorem 8
1.7 Outline of the Monograph 10
Part I: The Objects and the Tools 13
2 Polytopes and Realization Spaces 13
2.1 Notational Conventions 13
2.2 Polytopes, Cones and Combinatorial Polytopes 15
2.3 Affine and Projective Equivalence 18
2.4 Realization Spaces 19
2.5 Semialgebraic Sets and Stable Equivalence 21
2.6 Polarity 24
2.7 Visualization of 4 Polytopes: Schlegel Diagrams 26
3 Polytopal Constructions 28
3.1 PyTamids, Prisms and Tents 28
3.2 Connected Sums 29
3.3 Lawrence Extensions 32
3.4 Examples 37
Part II: The Universality Theorem 41
4 Equations and Polytopes 41
4.1 Shor s Normal Form 41
I
I
x CONTENTS ,
I
4.2 Encoding Equations into Polygons 42 l
5 The Basic Building Blocks 45
5.1 A Transmitter 46
5.2 The Connector 47 j
5.3 A Forgetful Transmitter 48 |
5.4 A 4 Polytope with Non Prescribable 2 Face 50 i
5.5 An Adapter •. 51 ;
5.6 A Polytope for Partial Transmission of Information .... 52 ¦
5.7 A Transmitter for Line Slopes 53
6 Harmonie Sets and Octagons 55
6.1 A Line Configuration Forcing Harmonie Relations .... 55
6.2 The Harmonie Polytope 56
7 Polytopes for Addition and Multiplication 59
7.1 Addition 59
7.2 Multiplication 62
8 Putting the Pieces Together: The Universality Theorem 68
8.1 Encoding Semialgebraic Sets in Polytopes 68
8.2 The Construction Seen frorn a Distance 69
8.3 Proving Stable Equivalence 71
Part III: Applications of Universality 77
9 Complexity Results 77
9.1 Algorithmic Complexity 77
9.2 Algebraic Complexity 78
9.3 The Sizes of 4 Polytopes 81
9.4 Infinite Classes of Non Polytopal Combinatorial 3 Spheres 84
10 Universality for 3 Diagrams and 4 Fans 87
10.1 3 Diagrams and 4 Fans 87
10.2 The Polytope P (S) 90
10.3 Nets 92
10.4 The Corollaries 95
11 The Universal Partition Theorem for 4 Polytopes 97
11.1 Semialgebraic Families and Partitions 97
11.2 Shor s Normal Form Versus Quadrilateral Sets 99
11.3 Computations of Polynomials 101
11.4 Encoding Quadrilateral Sets into Polytopes 106
11.5 The Switch Polytope 107
11.6 The Universal Partition Theorem 110
11.7 The Universal Partition Theorem for Point Configurations 112
CONTENTS xi
Part IV: Three dimensional Polytopes 117
12 Graphs 118
12.1 Preliminaries from Graph Theory 118
12.2 Tutte s Theorem on Stresses in Graphs 122
13 3 Polytopes 133
13.1 From Stressed Graphs to Polytopes 133
13.2 A Quantitative Analysis 140
13.3 The Structure of the Realization Space 144
Part V: Alternative Construction Techniques 149
14 Generalized Adapter Techniques 149
15 A Non Steinitz Theorem in Dimension Five 151
15.1 Conics and Incidence Theorems 151
15.2 An Incidence Theorem for 4 Polytopes 156
15.3 The Non Steinitz Theorem 160
16 The Universality Theorem in Dimension 6 162
16.1 Oriented Matroids 163
16.2 Zonotopes and Planets 165
16.3 The Construction 169
Part VI: Problems 173
17 Open Problems on Polytopes and Realization Spaces 173
17.1 Universality Theorems for Simplicial Polytopes 173
17.2 Small Non Rational 4 Polytopes 174
17.3 Many Polytopes 174
17.4 The Sizes of Polytopes 175
17.5 Rational Realizations of 3 Polytopes 176
17.6 The Steinitz Problem for Triangulated Tori 176
Bibliography 179
Index 183
|
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| illustrated | Illustrated |
| indexdate | 2024-07-09T18:03:28Z |
| institution | BVB |
| isbn | 3540620842 |
| language | German |
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| physical | XI, 187 S. graph. Darst. |
| publishDate | 1996 |
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| series2 | Lecture notes in mathematics |
| spelling | Richter-Gebert, Jürgen 1963- Verfasser (DE-588)111407346 aut Realization spaces of polytopes Jürgen Richter-Gebert Berlin [u.a.] Springer 1996 XI, 187 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Lecture notes in mathematics 1643 Teilw. zugl.: Berlin, Techn. Univ., Habil-Schr., 1995 Matroids Polytopes Polytop (DE-588)4175324-0 gnd rswk-swf (DE-588)4113937-9 Hochschulschrift gnd-content Polytop (DE-588)4175324-0 s DE-604 Lecture notes in mathematics 1643 (DE-604)BV000676446 1643 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=007413965&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Richter-Gebert, Jürgen 1963- Realization spaces of polytopes Lecture notes in mathematics Matroids Polytopes Polytop (DE-588)4175324-0 gnd |
| subject_GND | (DE-588)4175324-0 (DE-588)4113937-9 |
| title | Realization spaces of polytopes |
| title_auth | Realization spaces of polytopes |
| title_exact_search | Realization spaces of polytopes |
| title_full | Realization spaces of polytopes Jürgen Richter-Gebert |
| title_fullStr | Realization spaces of polytopes Jürgen Richter-Gebert |
| title_full_unstemmed | Realization spaces of polytopes Jürgen Richter-Gebert |
| title_short | Realization spaces of polytopes |
| title_sort | realization spaces of polytopes |
| topic | Matroids Polytopes Polytop (DE-588)4175324-0 gnd |
| topic_facet | Matroids Polytopes Polytop Hochschulschrift |
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