Lectures on polytopes:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
New York [u.a.]
Springer
2007
|
| Edition: | [corr. and] updated 7. print. |
| Series: | Graduate texts in mathematics
152 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Item Description: | Literaturverz. S. [325] - 366 |
| Physical Description: | XIII, 373 S. Ill., graph. Darst. |
| ISBN: | 9780387943657 9780387943299 |
Staff View
MARC
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| 245 | 1 | 0 | |a Lectures on polytopes |c Günter M. Ziegler |
| 250 | |a [corr. and] updated 7. print. | ||
| 264 | 1 | |a New York [u.a.] |b Springer |c 2007 | |
| 300 | |a XIII, 373 S. |b Ill., graph. Darst. | ||
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| 337 | |b n |2 rdamedia | ||
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| 490 | 1 | |a Graduate texts in mathematics |v 152 | |
| 500 | |a Literaturverz. S. [325] - 366 | ||
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Record in the Search Index
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| adam_text | Contents
Preface
v
Preface
to the Second Printing
...................
vii
Preface to the Seventh Printing
................... ix
0
Introduction and Examples
1
Notes
................................. 22
Problems and Exercises
....................... 23
1
Polytopes, Polyhedra, and Cones
27
1.1
The Main Theorem
..................... 27
1.2
Fourier-Motzkin Elimination: An
Affine
Sketch
....... 32
1.3
Fourier-Motzkin Elimination for Cones
............ 37
1.4
The
Farkas
Lemma
....................... 39
1.5
Recession Cone and Homogenization
............. 43
1.6
Carathéodory s
Theorem
.................... 45
Notes
................................. 47
Problems and Exercises
....................... 49
2
Faces of Polytopes
51
2.1
Vertices, Faces, and Facets
................... 51
2.2
The Face Lattice
........................ 55
2.3
Polarity
............................. 59
2.4
The Representation Theorem for Polytopes
......... 64
2.5
Simplicial and Simple Polytopes
............... 65
Contents
2.6 Appendix:
Projective
Transformations
............ 67
Notes ................................. 69
Problems and
Exercises
....................... 70
Graphs of Polytopes
77
3.1 Lines and Linear
Functions in
General Position....... 77
3.2
Directing the Edges
( Linear Programming
for Geometers )
80
3.3 The Hirsch
Conjecture
..................... 83
3.4
Kalai s Simple Way to Tell a Simple Polytope from Its Graph
93
3.5
Balinski s Theorem: The Graph is d-Connected
....... 95
Notes
................................. 96
Problems and Exercises
....................... 97
Steinitz Theorem for 3-Polytopes
103
4.1
S-Connected Planar Graphs
.................. 104
4.2
Simple
ΔΥ
Transformations Preserve Realizability
..... 107
4.3
Planar Graphs are AY Reducible
............... 109
4.4
Extensions of Steinitz Theorem
................ 113
Notes
................................. 115
Problems and Exercises
....................... 119
Schlegel
Diagrams for 4-Polytopes
127
5.1
Polyhedral Complexes
..................... 127
5.2 Schlegel
Diagrams
....................... 132
5.3
d-Diagrams
........................... 138
5.4
Three Examples
........................ 139
Notes
................................. 143
Problems and Exercises
....................... 145
Duality, Gale Diagrams, and Applications
149
6.1
Circuits and Cocircuits
..................... 150
(a)
Affine
Dependences
..................... 150
(b)
Affine
Functions
...................... 153
6.2
Vector Configurations
..................... 156
6.3
Oriented Matroids
....................... 157
(a) Axiomatics
............... . 159
(b) Equivalence
......................... 160
(c) Duality
...........................
163
(d) Deletion and Contraction
................. 163
6.4
Dual Configurations and Gale Diagrams
........... 165
6.5
Polytopes with Few Vertices
.................. 171
(a) A
Nonrational 8-Polytope................. 172
(b) Facets of 4-Polytopes Cannot be Prescribed
....... 173
(c) 2-Faces of 5-Polytopes Cannot be Prescribed
...... 175
(d) Polytopes Violating the
Isotopy
Conjecture
....... 177
Contents xiii
6.6
Rigidity and Universality
................... 179
Notes
................................. 183
Problems and Exercises
....................... 184
Fans, Arrangements, Zonotopes,
and Tilings
191
7.1
Fans
............................... 191
7.2
Projections and Minkowski Sums
............... 195
7.3
Zonotopes
............................ 198
7.4
Nonrealizable Oriented Matroids
............... 208
7.5
Zonotopal Tilings
........................ 217
Notes
................................. 224
Problems and Exercises
....................... 225
Shellability and the Upper Bound Theorem
231
8.1
Shellable and Nonshellable Complexes
............ 232
8.2
Shelling Polytopes
....................... 239
8.3
h-Vectors and Derm-Sommerville Equations
......... 246
8.4
The Upper Bound Theorem
.................. 254
8.5
Some Extremal Set Theory
.................. 258
8.6
The g-Theorem and Its Consequences
............ 268
Notes
................................. 275
Problems and Exercises
....................... 281
Fiber Polytopes, and Beyond
291
9.1
Polyhedral Subdivisions and Fiber Polytopes
........ 292
9.2
Some Examples
......................... 299
9.3
Constructing the Permuto-Associahedron
.......... 310
9.4
Toward a Category of Polytopes
?............... 319
Notes
................................. 320
Problems and Exercises
....................... 321
References
325
Index
367
|
| adam_txt |
Contents
Preface
v
Preface
to the Second Printing
.
vii
Preface to the Seventh Printing
. ix
0
Introduction and Examples
1
Notes
. 22
Problems and Exercises
. 23
1
Polytopes, Polyhedra, and Cones
27
1.1
The "Main Theorem"
. 27
1.2
Fourier-Motzkin Elimination: An
Affine
Sketch
. 32
1.3
Fourier-Motzkin Elimination for Cones
. 37
1.4
The
Farkas
Lemma
. 39
1.5
Recession Cone and Homogenization
. 43
1.6
Carathéodory's
Theorem
. 45
Notes
. 47
Problems and Exercises
. 49
2
Faces of Polytopes
51
2.1
Vertices, Faces, and Facets
. 51
2.2
The Face Lattice
. 55
2.3
Polarity
. 59
2.4
The Representation Theorem for Polytopes
. 64
2.5
Simplicial and Simple Polytopes
. 65
Contents
2.6 Appendix:
Projective
Transformations
. 67
Notes . 69
Problems and
Exercises
. 70
Graphs of Polytopes
77
3.1 Lines and Linear
Functions in
General Position. 77
3.2
Directing the Edges
("Linear Programming
for Geometers")
80
3.3 The Hirsch
Conjecture
. 83
3.4
Kalai's Simple Way to Tell a Simple Polytope from Its Graph
93
3.5
Balinski's Theorem: The Graph is d-Connected
. 95
Notes
. 96
Problems and Exercises
. 97
Steinitz' Theorem for 3-Polytopes
103
4.1
S-Connected Planar Graphs
. 104
4.2
Simple
ΔΥ
Transformations Preserve Realizability
. 107
4.3
Planar Graphs are AY Reducible
. 109
4.4
Extensions of Steinitz' Theorem
. 113
Notes
. 115
Problems and Exercises
. 119
Schlegel
Diagrams for 4-Polytopes
127
5.1
Polyhedral Complexes
. 127
5.2 Schlegel
Diagrams
. 132
5.3
d-Diagrams
. 138
5.4
Three Examples
. 139
Notes
. 143
Problems and Exercises
. 145
Duality, Gale Diagrams, and Applications
149
6.1
Circuits and Cocircuits
. 150
(a)
Affine
Dependences
. 150
(b)
Affine
Functions
. 153
6.2
Vector Configurations
. 156
6.3
Oriented Matroids
. 157
(a) Axiomatics
. . 159
(b) Equivalence
. 160
(c) Duality
.
163
(d) Deletion and Contraction
. 163
6.4
Dual Configurations and Gale Diagrams
. 165
6.5
Polytopes with Few Vertices
. 171
(a) A
Nonrational 8-Polytope. 172
(b) Facets of 4-Polytopes Cannot be Prescribed
. 173
(c) 2-Faces of 5-Polytopes Cannot be Prescribed
. 175
(d) Polytopes Violating the
Isotopy
Conjecture
. 177
Contents xiii
6.6
Rigidity and Universality
. 179
Notes
. 183
Problems and Exercises
. 184
Fans, Arrangements, Zonotopes,
and Tilings
191
7.1
Fans
. 191
7.2
Projections and Minkowski Sums
. 195
7.3
Zonotopes
. 198
7.4
Nonrealizable Oriented Matroids
. 208
7.5
Zonotopal Tilings
. 217
Notes
. 224
Problems and Exercises
. 225
Shellability and the Upper Bound Theorem
231
8.1
Shellable and Nonshellable Complexes
. 232
8.2
Shelling Polytopes
. 239
8.3
h-Vectors and Derm-Sommerville Equations
. 246
8.4
The Upper Bound Theorem
. 254
8.5
Some Extremal Set Theory
. 258
8.6
The g-Theorem and Its Consequences
. 268
Notes
. 275
Problems and Exercises
. 281
Fiber Polytopes, and Beyond
291
9.1
Polyhedral Subdivisions and Fiber Polytopes
. 292
9.2
Some Examples
. 299
9.3
Constructing the Permuto-Associahedron
. 310
9.4
Toward a Category of Polytopes
?. 319
Notes
. 320
Problems and Exercises
. 321
References
325
Index
367 |
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| dewey-full | 516.3/5 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 516 - Geometry |
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| dewey-search | 516.3/5 |
| dewey-sort | 3516.3 15 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
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| edition | [corr. and] updated 7. print. |
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| index_date | 2024-07-02T17:25:05Z |
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| spelling | Ziegler, Günter M. 1963- Verfasser (DE-588)121062155 aut Lectures on polytopes Günter M. Ziegler [corr. and] updated 7. print. New York [u.a.] Springer 2007 XIII, 373 S. Ill., graph. Darst. txt rdacontent n rdamedia nc rdacarrier Graduate texts in mathematics 152 Literaturverz. S. [325] - 366 Polytop Polytop (DE-588)4175324-0 gnd rswk-swf (DE-588)4143389-0 Aufgabensammlung gnd-content Aufgabensammlung Polytop (DE-588)4175324-0 s DE-604 Erscheint auch als Online-Ausgabe Ziegler, Günter M., 1963- Lectures on polytopes 978-1-4613-8431-1 Graduate texts in mathematics 152 (DE-604)BV000000067 152 Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015627688&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Ziegler, Günter M. 1963- Lectures on polytopes Graduate texts in mathematics Polytop Polytop (DE-588)4175324-0 gnd |
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| title | Lectures on polytopes |
| title_auth | Lectures on polytopes |
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