Convex analysis

The book showcases convexity in the context of mathematical analysis. It introduces analytic tools for studying convexity and provides analytical applications of the concept. The book includes a general background in 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 a thorough glossary to help readers with unfamiliar terms. Why Convexity?Basic IdeasIntroductionThe Classical TheorySeparation TheoremsApproximationFunctionsDefining FunctionAnalytic DefinitionConvex FunctionsExhaustion FunctionsMore on FunctionsOther CharacterizationsConvexity of Finite OrderExtreme PointsSupport FunctionsApproximation from BelowBumpingApplicationsThe Krein-Milman TheoremThe Minkowski SumBrunn-MinkowskiMore Sophisticated IdeasThe Polar of a SetOptimizationIntroductory ThoughtsSetup for the Simplex MethodAugmented FormThe Simplex AlgorithmGeneralizationsIntegral RepresentationThe Gamma FunctionHard Analytic FactsSums and ProjectionsThe MiniMax Theorem Concluding Remarks Appendix: Technical Tools Table of Notation Glossary 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 g Front Cover 1 Dedication 8 Table of Contents 10 Preface 12 Biography of Steven G. Krantz 14 Chapter 0: Why Convexity? 16 Chapter 1: Basic Ideas 18 Chapter 2: Characterization of Convexity Using Functions 36 Chapter 3: Further Developments Using Functions 72 Chapter 4: Applications of the Idea of Convexity 92 Chapter 5: More Sophisticated Ideas 106 Chapter 6: The MiniMax Theorem 148 Chapter 7: Concluding Remarks 154 Appendix: Technical Tools 156 Table of Notation 160 Glossary 162 Bibliography 170