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.
General
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