Lectures on Convex Sets

Lectures on Convex Sets

Valeriu Soltan


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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.

  • Prerequisites
  • The Affine Structure of ℝn
  • Convex Sets
  • Convex Hulls
  • Cones and Conic Hulls
  • Recession Directions
  • Closedness Conditions
  • Cone Polarity and Normal Directions
  • Bounds, Supports, and Asymptotes
  • Separation Properties
  • Extreme Structure of Convex Sets
  • Exposed Structure of Convex Sets
  • Polyhedra
  • Solutions to Problems

Readership: Undergraduate and graduate students in mathematics, optimization and operations research. Convex Set;Convex Hull;Cone;Support;Separation;Extreme;Exposed;Polyhedron0Key Features:
  • The exposition is self-contained, detailed and provides multiple cross-references, that makes the book accessible to a large audience
  • An essential part of the text is adapted from various research articles, never presented before in a textbook format
  • The book has a multidisciplinary nature: it can be useful to specialists in geometry, convex analysis, operations research, and optimization
  • The new edition contains new chapters and additional exercises with respective solutions
  • Despite the presence of a large number of monographs on convex sets, there are quite a few textbooks on this topic. This book is to the level of graduate study, with higher degree of complexity and essentially more research-related results and references