Theory of Convex Structures (eBook)

M.L.J. van de Vel (Autor)

eBook Download: PDF

1993 | 1. Auflage
539 Seiten
Elsevier Science (Verlag)
978-0-08-093310-8 (ISBN)

Infos

Presented in this monograph is the current state-of-the-art in the theory of convex structures. The notion of convexity covered here is considerably broader than the classic one, specifically, it is not restricted to the context of vector spaces. Classical concepts of order-convex sets (Birkhoff) and of geodesically convex sets (Menger) are directly inspired by intuition, they go back to the first half of this century. An axiomatic approach started to develop in the early Fifties. The author became attracted to it in the mid-Seventies, resulting in the present volume, in which graphs appear side-by-side with Banach spaces, classical geometry with matroids, and ordered sets with metric spaces. A wide variety of results has been included (ranging for instance from the area of partition calculus to that of continuous selection). The tools involved are borrowed from areas ranging from discrete mathematics to infinite-dimensional topology.

Although addressed primarily to the researcher, parts of this monograph can be used as a basis for a well-balanced, one-semester graduate course.

Inhalt

Front Cover 1
Theory of Convex Structures 4
Copyright Page 5
Introduction 8
Table of Contents 14
List of frequent Symbols 16
Chapter I Abstract Convex Structures 17
§1. Basic Concepts 19
§2. The Hull Operator 47
§3. Half-spaces and Separation 65
§4. Interval Spaces 87
§5. Base-point Orders 107
§6. Modular Spaces 129
§7. Bryant-Webster Spaces 159
Chapter II Convex Invariants 175
§1. Classical Convex Invariants 177
§2. Invariants and Product Spaces 197
§3. Invariants in other Constructions 221
§4. Infinite Combinatorics 235
§5. Tverberg Numbers 267
Chapter III Topological Convex Structures 281
§1. Topology and Convexity on the same Set 283
§2. Continuity of the Hull Operator 303
§3. Uniform Convex Structures 319
§4. Top-convex Separation 341
§5. Intrinsic Topology 365
Chapter IV Miscellaneous 393
§1. Embedding Bryant-Webster Spaces into Vector Spaces 395
§2. Extremality, Pseudo-boundary and Pseudo-interior 421
§3. Continuous Selection 451
§4. Dimension Theory 469
§5. Dimension and Convex Invariants 485
§6. Fixed Points 503
Bibliography 523
Index of Terms 545

Produkt-Details

• Erscheint lt. Verlag: 2.8.1993
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik ► Algebra
• Themenwelt: Mathematik / Informatik ► Mathematik ► Geometrie / Topologie
• Themenwelt: Mathematik / Informatik ► Mathematik ► Graphentheorie
• Themenwelt: Technik
• ISBN-10: 0-08-093310-6 / 0080933106
• ISBN-13: 978-0-08-093310-8 / 9780080933108