**Graduate Studies in Mathematics**

Volume: 54;
2002;
366 pp;
Hardcover

MSC: Primary 52; 46; 90; 49;

Print ISBN: 978-0-8218-2968-4

Product Code: GSM/54

List Price: $72.00

Individual Member Price: $57.60

**Electronic ISBN: 978-1-4704-1792-5
Product Code: GSM/54.E**

List Price: $72.00

Individual Member Price: $57.60

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# A Course in Convexity

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*Alexander Barvinok*

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 & Endorsements

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

# Table of Contents

## A Course in Convexity

- Cover Cover11 free
- Title page i2 free
- Contents iii4 free
- Preface vii8 free
- Convex sets at large 112 free
- Faces and extreme points 4152
- Convex sets in topological vector spaces 105116
- Polarity, duality and linear programming 143154
- Convex bodies and ellipsoids 203214
- Faces of polytopes 249260
- Lattices and convex bodies 279290
- Lattice points and polyhedra 325336
- Bibliography 357368
- Index 363374 free
- Back Cover Back Cover1378