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A Course in Convexity
Alexander Barvinok, University of Michigan, Ann Arbor, MI
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Graduate Studies in Mathematics
2002; 366 pp; hardcover
Volume: 54
ISBN-10: 0-8218-2968-8
ISBN-13: 978-0-8218-2968-4
List Price: US$68
Member Price: US$54.40
Order Code: GSM/54
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Convexity is a simple idea that manifests itself in a surprising variety of places. This fertile field has an immensely rich structure and numerous applications. Barvinok demonstrates that simplicity, intuitive appeal, and the universality of applications make teaching (and learning) convexity a gratifying experience. The book will benefit both teacher and student: It is easy to understand, entertaining to the reader, and includes many exercises that vary in degree of difficulty. Overall, the author demonstrates the power of a few simple unifying principles in a variety of pure and applied problems.

The notion of convexity comes from geometry. Barvinok describes here its geometric aspects, yet he focuses on applications of convexity rather than on convexity for its own sake. Mathematical applications range from analysis and probability to algebra to combinatorics to number theory. Several important areas are covered, including topological vector spaces, linear programming, ellipsoids, and lattices. Specific topics of note are optimal control, sphere packings, rational approximations, numerical integration, graph theory, and more. And of course, there is much to say about applying convexity theory to the study of faces of polytopes, lattices and polyhedra, and lattices and convex bodies.

The prerequisites are minimal amounts of linear algebra, analysis, and elementary topology, plus basic computer skills. Portions of the book could be used by advanced undergraduates. As a whole, it is designed for graduate students interested in mathematical methods, computer science, electrical engineering, and operations research. Readers will find some new results. Also, many known results are discussed from a new perspective.

Readership

Advanced undergraduates, graduate students, and researchers interested in mathematical methods, computer science, electrical engineering, and operations research.

Reviews

"An excellent choice of textbook for a geometry course ... Everything the reader needs is defined in the book ... The chapters are well integrated ... I enthusiastically recommend [the book]. It effectively demonstrates how convexity connects with just about all branches of mathematics. The book is well illustrated and well written ... In reading it, I get the sense of how enjoyable it would be to hear Barvinok lecture on the material. I hope that it will attract many students to this branch of geometry."

-- MAA Monthly

"My impression is that the book would be fine to teach from ... it contains many useful diagrams. The test is well written, and everything is clearly explained ... wealth of material that it contains and the excellence of its treatment would make this book a desirable addition to one's library. I recommend it highly."

-- Bulletin of the LMS

Table of Contents

  • Convex sets at large
  • Faces and extreme points
  • Convex sets in topological vector spaces
  • Polarity, duality and linear programming
  • Convex bodies and ellipsoids
  • Faces of polytopes
  • Lattices and convex bodies
  • Lattice points and polyhedra
  • Bibliography
  • Index
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