This book provides a systematic treatment of algebraic and topological properties of convex sets (possibly non-closed or unbounded) in the n-dimensional Euclidean space. Topics under consideration include general properties of convex sets and convex hulls, cones and conic hulls, polyhedral sets, the extreme structure, support and separation properties of convex sets.
Lectures on Convex Sets is self-contained and unified in presentation. The book grew up out of various courses on geometry and convexity, taught by the author for more than a decade. It can be used as a textbook for graduate students and even ambitious undergraduates in mathematics, optimization, and operations research. It may also be viewed as a supplementary book for a course on convex geometry or convex analysis, or as a source for independent study of the subject, suitable for non-geometers.
Sample Chapter(s)
Chapter 0: Prerequisites (326 KB)
Contents:
- The Affine Structure of ℝn
- Convex Sets
- Convex Hulls
- Convex Cones and Conic Hulls
- Recession and Normal Directions
- Support and Separation Properties
- The Extreme Structure of Convex Sets
- The Exposed Structure of Convex Sets
- Polyhedra
Readership: Graduate students in mathematics, optimization and operations research.
"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
