Convex Analysis  book cover
1st Edition

Convex Analysis

ISBN 9781498706377
Published October 20, 2014 by Chapman and Hall/CRC
176 Pages 64 B/W Illustrations

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Book Description

Convexity is an ancient idea going back to Archimedes. Used sporadically in the mathematical literature over the centuries, today it is a flourishing area of research and a mathematical subject in its own right. Convexity is used in optimization theory, functional analysis, complex analysis, and other parts of mathematics.

Convex Analysis introduces analytic tools for studying convexity and provides analytical applications of the concept. The book includes a general background on classical geometric theory which allows readers to obtain a glimpse of how modern mathematics is developed and how geometric ideas may be studied analytically.

Featuring a user-friendly approach, the book contains copious examples and plenty of figures to illustrate the ideas presented. It also includes an appendix with the technical tools needed to understand certain arguments in the book, a tale of notation, and a thorough glossary to help readers with unfamiliar terms. This book is a definitive introductory text to the concept of convexity in the context of mathematical analysis and a suitable resource for students and faculty alike.

Table of Contents

Why Convexity?

Basic Ideas
The Classical Theory
Separation Theorems

Defining Function
Analytic Definition
Convex Functions
Exhaustion Functions

More on Functions
Other Characterizations
Convexity of Finite Order
Extreme Points
Support Functions
Approximation from Below

The Krein-Milman Theorem
The Minkowski Sum

More Sophisticated Ideas
The Polar of a Set
Introductory Thoughts
Setup for the Simplex Method
Augmented Form
The Simplex Algorithm
Integral Representation
The Gamma Function
Hard Analytic Facts
Sums and Projections

The MiniMax Theorem

Concluding Remarks

Appendix: Technical Tools

Table of Notation


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Steven G. Krantz received his PhD from Princeton University. Dr. Krantz has taught at UCLA, Princeton University, Penn State, and Washington University in St. Louis. For five years he was the department chair at Washington University. Dr. Krantz has received the Chauvenet Prize, awarded for outstanding mathematical expository writing. He has also been awarded the Kemper Prize and the Beckenbach Book Award. Dr. Krantz is the founding editor of the Journal of Geometric Analysis as well as Complex Analysis and its Synergies. He has written nearly 195 scholarly papers and more than 75 books.


"The goal of this user-friendly and self-contained book is to introduce and study the notion of convexity of sets and functions in Euclidean spaces with a perspective that is closer than usual to the approach and the tools normally prerogatives of the mathematical analysis setting."
Zentralblatt MATH 1310