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Cover: 9780387907222 | An Introduction to Convex Polytopes | Arne Brondsted | Buch | viii
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An Introduction to Convex Polytopes
Buch von Arne Brondsted
Sprache: Englisch

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Beschreibung
The aim of this book is to introduce the reader to the fascinating world of convex polytopes. The highlights of the book are three main theorems in the combinatorial theory of convex polytopes, known as the Dehn-Sommerville Relations, the Upper Bound Theorem and the Lower Bound Theorem. All the background information on convex sets and convex polytopes which is m~eded to under­ stand and appreciate these three theorems is developed in detail. This background material also forms a basis for studying other aspects of polytope theory. The Dehn-Sommerville Relations are classical, whereas the proofs of the Upper Bound Theorem and the Lower Bound Theorem are of more recent date: they were found in the early 1970's by P. McMullen and D. Barnette, respectively. A famous conjecture of P. McMullen on the charac­ terization off-vectors of simplicial or simple polytopes dates from the same period; the book ends with a brief discussion of this conjecture and some of its relations to the Dehn-Sommerville Relations, the Upper Bound Theorem and the Lower Bound Theorem. However, the recent proofs that McMullen's conditions are both sufficient (L. J. Billera and C. W. Lee, 1980) and necessary (R. P. Stanley, 1980) go beyond the scope of the book. Prerequisites for reading the book are modest: standard linear algebra and elementary point set topology in [R1d will suffice.
The aim of this book is to introduce the reader to the fascinating world of convex polytopes. The highlights of the book are three main theorems in the combinatorial theory of convex polytopes, known as the Dehn-Sommerville Relations, the Upper Bound Theorem and the Lower Bound Theorem. All the background information on convex sets and convex polytopes which is m~eded to under­ stand and appreciate these three theorems is developed in detail. This background material also forms a basis for studying other aspects of polytope theory. The Dehn-Sommerville Relations are classical, whereas the proofs of the Upper Bound Theorem and the Lower Bound Theorem are of more recent date: they were found in the early 1970's by P. McMullen and D. Barnette, respectively. A famous conjecture of P. McMullen on the charac­ terization off-vectors of simplicial or simple polytopes dates from the same period; the book ends with a brief discussion of this conjecture and some of its relations to the Dehn-Sommerville Relations, the Upper Bound Theorem and the Lower Bound Theorem. However, the recent proofs that McMullen's conditions are both sufficient (L. J. Billera and C. W. Lee, 1980) and necessary (R. P. Stanley, 1980) go beyond the scope of the book. Prerequisites for reading the book are modest: standard linear algebra and elementary point set topology in [R1d will suffice.
Inhaltsverzeichnis
1 Convex Sets.- A7;1. The Affine Structure of ?d.- A7;2. Convex Sets.- A7;3. The Relative Interior of a Convex Set.- A7;4. Supporting Hyperplanes and Halfspaces.- A7;5. The Facial Structure of a Closed Convex Set.- A7;6. Polarity.- 2 Convex Polytopes.- A7;7. Polytopes.- A7;8. Polyhedral Sets.- A7;9. Polarity of Polytopes and Polyhedral Sets.- A7;10. Equivalence and Duality of Polytopes.- A7;11. Vertex-Figures.- A7;12. Simple and Simplicial Polytopes.- A7;13. Cyclic Polytopes.- A7;14. Neighbourly Polytopes.- A7;15. The Graph of a Polytope.- 3 Combinatorial Theory of Convex Polytopes.- A7;16. Eulerߣs Relation.- A7;17. The Dehn-Sommerville Relations.- A7;18. The Upper Bound Theorem.- A7;19. The Lower Bound Theorem.- A7;20. McMullenߣs Conditions.- Appendix 1 Lattices.- Appendix 2 Graphs.- Appendix 3 Combinatorial Identities.- Bibliographical Comments.- List of Symbols.
Details
Erscheinungsjahr: 1982
Fachbereich: Wahrscheinlichkeitstheorie
Genre: Mathematik
Rubrik: Naturwissenschaften & Technik
Medium: Buch
Seiten: 176
Reihe: Graduate Texts in Mathematics
Inhalt: viii
162 S.
ISBN-13: 9780387907222
ISBN-10: 038790722X
Sprache: Englisch
Ausstattung / Beilage: HC runder Rücken kaschiert
Einband: Gebunden
Autor: Brondsted, Arne
Auflage: 1983
Hersteller: Springer US
Springer New York
Graduate Texts in Mathematics
Maße: 241 x 160 x 15 mm
Von/Mit: Arne Brondsted
Erscheinungsdatum: 21.12.1982
Gewicht: 0,436 kg
preigu-id: 102424800
Inhaltsverzeichnis
1 Convex Sets.- A7;1. The Affine Structure of ?d.- A7;2. Convex Sets.- A7;3. The Relative Interior of a Convex Set.- A7;4. Supporting Hyperplanes and Halfspaces.- A7;5. The Facial Structure of a Closed Convex Set.- A7;6. Polarity.- 2 Convex Polytopes.- A7;7. Polytopes.- A7;8. Polyhedral Sets.- A7;9. Polarity of Polytopes and Polyhedral Sets.- A7;10. Equivalence and Duality of Polytopes.- A7;11. Vertex-Figures.- A7;12. Simple and Simplicial Polytopes.- A7;13. Cyclic Polytopes.- A7;14. Neighbourly Polytopes.- A7;15. The Graph of a Polytope.- 3 Combinatorial Theory of Convex Polytopes.- A7;16. Eulerߣs Relation.- A7;17. The Dehn-Sommerville Relations.- A7;18. The Upper Bound Theorem.- A7;19. The Lower Bound Theorem.- A7;20. McMullenߣs Conditions.- Appendix 1 Lattices.- Appendix 2 Graphs.- Appendix 3 Combinatorial Identities.- Bibliographical Comments.- List of Symbols.
Details
Erscheinungsjahr: 1982
Fachbereich: Wahrscheinlichkeitstheorie
Genre: Mathematik
Rubrik: Naturwissenschaften & Technik
Medium: Buch
Seiten: 176
Reihe: Graduate Texts in Mathematics
Inhalt: viii
162 S.
ISBN-13: 9780387907222
ISBN-10: 038790722X
Sprache: Englisch
Ausstattung / Beilage: HC runder Rücken kaschiert
Einband: Gebunden
Autor: Brondsted, Arne
Auflage: 1983
Hersteller: Springer US
Springer New York
Graduate Texts in Mathematics
Maße: 241 x 160 x 15 mm
Von/Mit: Arne Brondsted
Erscheinungsdatum: 21.12.1982
Gewicht: 0,436 kg
preigu-id: 102424800
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