Constantin P. Niculescu - Böcker

This second edition provides a thorough introduction to contemporary convex function theory with many new results. A large variety of subjects are covered, from the one real variable case to some of the most advanced topics.  The new edition includes considerably more material emphasizing the rich applicability of convex analysis to concrete examples.  Chapters 4, 5, and 6 are entirely new, covering important topics such as the Hardy-Littlewood-Pólya-Schur theory of majorization, matrix convexity, and the Legendre-Fenchel-Moreau duality theory.This book can serve as a reference and source of inspiration to researchers in several branches of mathematics and engineering, and it can also be used as a reference text for graduate courses on convex functions and applications.

Mainly addressed to researchers in the fields of generalized measure theory, nonlinear integrals, nonlinear functional analysis, and nonlinear approximation theory, this book is based mainly on the authors’ results. While the Choquet integral has proven to be powerful and useful in cooperative games, decision theory, finance, economics, insurance, pattern recognition, artificial intelligence, and automated reasoning, these applications are not the focus in this monograph. Instead, the special properties of Choquet’s integral, inspired the study of various aspects for the class of monotone sublinear operators (more general than the positive linear operators) on Banach ordered spaces, as for example Korovkin-type results. The main ideas in this book also include the replacement in various known mathematical concepts, of the classical linear Lebesgue integral by the nonlinear (sublinear and monotone)  Choquet integral and to study the new obtained concepts.

This third edition presents an expanded and updated treatment of convex analysis methods, incorporating many new results that have emerged in recent years.

This third edition presents an expanded and updated treatment of convex analysis methods, incorporating many new results that have emerged in recent years. These additions are essential for grasping the practical applications of convex function theory in solving contemporary real-world problems.To reflect these advancements, the material has been meticulously reorganized, with a greater emphasis on topics relevant to current research. Additionally, great care has been taken to ensure that the text remains accessible to a broad audience, including both students and researchers focused on the application of mathematics.Ideal for undergraduate courses, graduate seminars, or as a comprehensive reference, this book is an indispensable resource for those seeking to understand the extensive potential of convex function theory.

The book targets undergraduate and postgraduate mathematics students and helps them develop a deep understanding of mathematical analysis.