Lectures On Convex Sets (Second Edition)

Exceeds maximum length of 2000 characters; Exceeds maximum length of 2000 characters; Review of the First Edition: "The book is carefully written and should be useful as a textbook for courses on convex geometry … Also of interest are the notes after each chapter. More of these would be appreciated by an expert reader. The exercises are a treasure. The book serves its purpose well and can be recommended without reservation." Mathematical Reviews Clippings "This is an extremely complete book on the fundamentals of the algebraic and topological properties of convex sets. This is a well-written book, which should be of considerable use to researchers in convex geometry." Zentralblatt MATH "The book is very well written. In spite of its relatively elementary level, the book contains many important results in finite dimensional convexity, necessary in many other mathematical areas. By the detailed and rigorous presentation of the material it can be recommended for self-study as well." Studia Universitatis Babeş-Bolyai Mathematica 5-star Reviews from Amazon: "Convex analysis is of extreme importance in mathematics, especially in optimization. Fortunately, there are numerous excellent books on this subject; this new one is equally outstanding. Soltan’s text has detailed coverage of the algebraic and geometric properties of subspaces, affine sets, convex sets, and cones. Of special note is the extensive treatment of various types of cones, including recession, normal, polar, and barrier. Topological properties of the closure, interior, relative interior, boundary, and relative boundary are extensively covered along with separation properties and extreme structure. Many of the topics, such as the apex set of a cone, seem to be unique to this book. Also unique is the presentation of numerous results in terms of the Minkowski (element-wise) sum of sets. The notation and terminology are clear and precise, and complete proofs are given. Of special merit is numerous counterexamples showing that various strengthenings are false.. This book provides an outstanding treatment of convex analysis that is suitable for self-study, and it is clearly a valuable addition to the existing literature." dsbreads Amazon.com Reviewer The book provides a self-contained and systematic treatment of algebraic and topological properties of convex sets in the n-dimensional Euclidean space. It benefits advanced undergraduate and graduate students with various majors in mathematics, optimization, and operations research. It may be adapted as a primary book or an additional text for any course in convex geometry or convex analysis, aimed at non-geometers. It can be a source for independent study and a reference book for researchers in academia. The second edition essentially extends and revises the original book. Every chapter is rewritten, with many new theorems, examples, problems, and bibliographical references included. It contains three new chapters and 100 additional problems with solutions.