Convex Sets and Their Applications

Convex Sets and Their Applications

by Steven R. Lay

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Overview

Suitable for advanced undergraduates and graduate students, this text introduces the broad scope of convexity. It leads students to open questions and unsolved problems, and it highlights diverse applications. Author Steven R. Lay, Professor of Mathematics at Lee University in Tennessee, reinforces his teachings with numerous examples, plus exercises with hints and answers.

The first three chapters form the foundation for all that follows, starting with a review of the fundamentals of linear algebra and topology. They also survey the development and applications of relationships between hyperplanes and convex sets. Subsequent chapters are relatively self-contained, each focusing on a particular aspect or application of convex sets. Topics include characterizations of convex sets, polytopes, duality, optimization, and convex functions. Hints, solutions, and references for the exercises appear at the back of the book.

Product Details

ISBN-13: 9780486458038
Publisher: Dover Publications
Publication date: 06/05/2007
Series: Dover Books on Mathematics Series
Pages: 256
Product dimensions: 5.37(w) x 8.50(h) x (d)

Table of Contents


Fundamentals     1
Linear Algebra and Topology     1
Convex Sets     10
Hyperplanes     27
Hyperplanes and Linear Functionals     27
Separating Hyperplanes     33
Supporting Hyperplanes     41
Helly-Type Theorems     47
Helly's Theorem     47
Kirchberger's Theorem     55
Kirchberger-type Theorems     61
Separation by a Spherical Surface     61
Separation by a Cylinder     64
Separation by a Parallelotope     70
Special Topics in E[superscript 2]     76
Sets of Constant Width     76
Universal Covers     84
The Isoperimetric Problem     88
Families of Convex Sets     94
Parallel Bodies     94
The Blaschke Selection Theorem     97
The Existence of Extremal Sets     101
Characterizations of Convex Sets     104
Local Convexity     104
Local Support Properties     107
Nearest-Point Properties     111
Polytopes     116
The Faces of a Polytope     116
Special Types of Polytopes and Euler's Formula     123
Approximation by Polytopes     133
Duality     140
Polarity and Polytopes     140
Dual Cones     146
Optimization     154
Finite Matrix Games     154
Linear Programming     168
The Simplex Method     183
Convex Functions     158
Basic Properties     198
Support and Distance Functions     205
Continuity and Differentiability     214
Solutions, Hints, and References for Exercises     222
Bibliography     234
Index     239

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