A course in convexity:
"Barvinok demonstrates that simplicity, intuitive appeal, and the universality of applications make teaching (and learning) convexity a gratifying experience. The prerequisites are minimal amounts of linear algebra, analysis, and elementary topology, plus basic computational skills. Portions of...
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Providence, RI
American Mathematical Society
2002
|
| Series: | Graduate studies in mathematics
54 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis Klappentext |
| Summary: | "Barvinok demonstrates that simplicity, intuitive appeal, and the universality of applications make teaching (and learning) convexity a gratifying experience. The prerequisites are minimal amounts of linear algebra, analysis, and elementary topology, plus basic computational skills. Portions of the book could be used by advanced undergraduates. As a whole, it is designed for graduate students interested in mathematical methods, computer science, electrical engineering, and operations research. The book will also be of interest to research mathematicians, who will find some results that are recent, some that are new, and many known results that are discussed from a new perspective."--BOOK JACKET. |
| Physical Description: | X, 366 S. graph. Darst. |
| ISBN: | 0821829688 9780821829684 |
Staff View
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Record in the Search Index
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| adam_text | Contents
Preface vii
Chapter I. Convex Sets at Large 1
1. Convex Sets. Main Definitions, Some Interesting Examples
and Problems 1
2. Properties of the Convex Hull. Caratheodory s Theorem 7
3. An Application: Positive Polynomials 12
4. Theorems of Radon and Helly 17
5. Applications of Helly s Theorem in Combinatorial Geome¬
try 21
6. An Application to Approximation 24
7. The Euler Characteristic 28
8. Application: Convex Sets and Linear Transformations 33
9. Polyhedra and Linear Transformations 37
10. Remarks 39
Chapter II. Faces and Extreme Points 41
1. The Isolation Theorem 41
2. Convex Sets in Euclidean Space 47
3. Extreme Points. The Krein Milman Theorem for Euclidean
Space 51
4. Extreme Points of Polyhedra 53
iii
iv Contents
5. The Birkhoff Polytope 56
6. The Permutation Polytope and the Schur Horn Theorem 58
7. The Transportation Polyhedron 60
8. Convex Cones 65
9. The Moment Curve and the Moment Cone 67
10. An Application: Double Precision Formulas for Numeri¬
cal Integration 70
11. The Cone of Non negative Polynomials 73
12. The Cone of Positive Semidefinite Matrices 78
13. Linear Equations in Positive Semidefinite Matrices 83
14. Applications: Quadratic Convexity Theorems 89
15. Applications: Problems of Graph Realizability 94
16. Closed Convex Sets 99
17. Remarks 103
Chapter III. Convex Sets in Topological Vector Spaces 105
1. Separation Theorems in Euclidean Space and Beyond 105
2. Topological Vector Spaces, Convex Sets and Hyperplanes 109
3. Separation Theorems in Topological Vector Spaces 117
4. The Krein Milman Theorem for Topological Vector Spaces 121
5. Polyhedra in L°° 123
6. An Application: Problems of Linear Optimal Control 126
7. An Application: The Lyapunov Convexity Theorem 130
8. The Simplex of Probability Measures 133
9. Extreme Points of the Intersection. Applications 136
10. Remarks 141
Chapter IV. Polarity, Duality and Linear Programming 143
1. Polarity in Euclidean Space 143
2. An Application: Recognizing Points in the Moment Cone 150
3. Duality of Vector Spaces 154
4. Duality of Topological Vector Spaces 157
5. Ordering a Vector Space by a Cone 160
6. Linear Programming Problems 162
7. Zero Duality Gap 166
8. Polyhedral Linear Programming 172
Contents v
9. An Application: The Transportation Problem 176
10. Semidefinite Programming 178
11. An Application: The Clique and Chromatic Numbers of a
Graph 182
12. Linear Programming in L°° 185
13. Uniform Approximation as a Linear Programming Problem 191
14. The Mass Transfer Problem 196
15. Remarks 202
Chapter V. Convex Bodies and Ellipsoids 203
1. Ellipsoids 203
2. The Maximum Volume Ellipsoid of a Convex Body 207
3. Norms and Their Approximations 216
4. The Ellipsoid Method 225
5. The Gaussian Measure on Euclidean Space 232
6. Applications to Low Rank Approximations of Matrices 240
7. The Measure and Metric on the Unit Sphere 244
8. Remarks 248
Chapter VI. Faces of Polytopes 249
1. Polytopes and Polarity 249
2. The Facial Structure of the Permutation Polytope 254
3. The Euler Poincare Formula 258
4. Polytopes with Many Faces: Cyclic Polytopes 262
5. Simple Polytopes 264
6. The /i vector of a Simple Polytope.
Derm Sommerville Equations 267
7. The Upper Bound Theorem 270
8. Centrally Symmetric Polytopes 274
9. Remarks 277
Chapter VII. Lattices and Convex Bodies 279
1. Lattices 279
2. The Determinant of a Lattice 286
3. Minkowski s Convex Body Theorem 293
vi Contents
4. Applications: Sums of Squares and Rational Approxima¬
tions 298
5. Sphere Packings 302
6. The Minkowski Hlawka Theorem 305
7. The Dual Lattice 309
8. The Flatness Theorem 315
9. Constructing a Short Vector and a Reduced Basis 319
10. Remarks 324
Chapter VIII. Lattice Points and Polyhedra 325
1. Generating Functions and Simple Rational Cones 325
2. Generating Functions and Rational Cones 330
3. Generating Functions and Rational Polyhedra 335
4. Brion s Theorem 341
5. The Ehrhart Polynomial of a Polytope 349
6. Example: Totally Unimodular Polytopes 353
7. Remarks 356
Bibliography 357
Index 363
Convexity is a simple idea that manifests itself in a surprising variety of places.
This fertile field has an immensely rich structure and numerous applications.
Barvinok demonstrates that simplicity, intuitive appeal, and the universality of
applications make teaching (and learning) convexity a gratifying experience.The
book will benefit both teacher and student: It is easy to understand, entertaining
to the reader, and includes many exercises that vary in degree of difficulty.
Overall, the author demonstrates the power of a few simple unifying principles
in a variety of pure and applied problems.
The prerequisites are minimal amounts of linear algebra, analysis, and elemen¬
tary topology, plus basic computational skills. Portions of the book could be
used by advanced undergraduates. As a whole, it is designed for graduate
students interested in mathematical methods, computer science, electrical engi¬
neering, and operations research.The book will also be of interest to research
mathematicians, who will find some results that are recent, some that are new,
and many known results that are discussed from a new perspective.
|
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| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| format | Book |
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| spelling | Barvinok, Alexander 1963- Verfasser (DE-588)137398034 aut A course in convexity Alexander Barvinok Providence, RI American Mathematical Society 2002 X, 366 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Graduate studies in mathematics 54 "Barvinok demonstrates that simplicity, intuitive appeal, and the universality of applications make teaching (and learning) convexity a gratifying experience. The prerequisites are minimal amounts of linear algebra, analysis, and elementary topology, plus basic computational skills. Portions of the book could be used by advanced undergraduates. As a whole, it is designed for graduate students interested in mathematical methods, computer science, electrical engineering, and operations research. The book will also be of interest to research mathematicians, who will find some results that are recent, some that are new, and many known results that are discussed from a new perspective."--BOOK JACKET. Analyse fonctionnelle Convexe functies gtt Convexe verzamelingen gtt Funções convexas (análise funcional) larpcal Géométrie convexe Programmation (Mathématiques) Convex geometry Functional analysis Programming (Mathematics) Konvexe Menge (DE-588)4165212-5 gnd rswk-swf Konvexe Geometrie (DE-588)4407260-0 gnd rswk-swf Konvexe Menge (DE-588)4165212-5 s DE-604 Konvexe Geometrie (DE-588)4407260-0 s Erscheint auch als Online-Ausgabe 978-1-4704-1792-5 Graduate studies in mathematics 54 (DE-604)BV009739289 54 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=009945322&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis Digitalisierung UB Passau - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=009945322&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA Klappentext |
| spellingShingle | Barvinok, Alexander 1963- A course in convexity Graduate studies in mathematics Analyse fonctionnelle Convexe functies gtt Convexe verzamelingen gtt Funções convexas (análise funcional) larpcal Géométrie convexe Programmation (Mathématiques) Convex geometry Functional analysis Programming (Mathematics) Konvexe Menge (DE-588)4165212-5 gnd Konvexe Geometrie (DE-588)4407260-0 gnd |
| subject_GND | (DE-588)4165212-5 (DE-588)4407260-0 |
| title | A course in convexity |
| title_auth | A course in convexity |
| title_exact_search | A course in convexity |
| title_full | A course in convexity Alexander Barvinok |
| title_fullStr | A course in convexity Alexander Barvinok |
| title_full_unstemmed | A course in convexity Alexander Barvinok |
| title_short | A course in convexity |
| title_sort | a course in convexity |
| topic | Analyse fonctionnelle Convexe functies gtt Convexe verzamelingen gtt Funções convexas (análise funcional) larpcal Géométrie convexe Programmation (Mathématiques) Convex geometry Functional analysis Programming (Mathematics) Konvexe Menge (DE-588)4165212-5 gnd Konvexe Geometrie (DE-588)4407260-0 gnd |
| topic_facet | Analyse fonctionnelle Convexe functies Convexe verzamelingen Funções convexas (análise funcional) Géométrie convexe Programmation (Mathématiques) Convex geometry Functional analysis Programming (Mathematics) Konvexe Menge Konvexe Geometrie |
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