Convex Sets and Their Applications

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ISBN: 9780486458038 | 0486458032
Cover: Paperback
Copyright: 6/5/2007

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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. Numerous examples, exercises, hints, and answers appear throughout the text. The author is Professor of Mathematics at Lee University in Tennessee.

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

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