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. <P>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. <P>The prerequisites are minimal amounts of linear algebra, analysis, and elementary topology, plus basic computational 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. The book will also be of interest to research mathematicians, who will find some results that are recent, some that are new, and many known results that are discussed from a new perspective.
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"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
"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
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Inside This Book
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First Sentence:
We define convex sets and explore some of their fundamental properties. Read the first page Key Phrases - Statistically Improbable Phrases (SIPs): (learn more)
permutation polytope, graph realizability, algebraically open set, simple rational cone, maximum volume ellipsoid, compact hase, unimodular cone, convex body symmetric, rotation invariant probability measure, integer polytope, rational polyhedron, rational polyhedra, algebraic interior, conic combination, flatness theorem, packing radius, simple polytope, fundamental parallelepiped, moment cone, rational polytope, rational cones, zero duality gap, positive semidefinite matrices, conic hull, affine hull Key Phrases - Capitalized Phrases (CAPs): (learn more)
Helly's Theorem, Carathéodory's Theorem, Brion's Theorem, Minkowski's Convex Body Theorem, Bipolar Theorem, Pick's Formula, Diet Problem, Applying Lemma, Radon's Theorem, Sketch of Proof, Minkowski's Theorem, Neumann Theorem, Dvoretzky's Theorem, Recognizing Points New!
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Front Cover | Table of Contents | First Pages | Index | Back Cover | Surprise Me!
We define convex sets and explore some of their fundamental properties. Read the first page Key Phrases - Statistically Improbable Phrases (SIPs): (learn more)
permutation polytope, graph realizability, algebraically open set, simple rational cone, maximum volume ellipsoid, compact hase, unimodular cone, convex body symmetric, rotation invariant probability measure, integer polytope, rational polyhedron, rational polyhedra, algebraic interior, conic combination, flatness theorem, packing radius, simple polytope, fundamental parallelepiped, moment cone, rational polytope, rational cones, zero duality gap, positive semidefinite matrices, conic hull, affine hull Key Phrases - Capitalized Phrases (CAPs): (learn more)
Helly's Theorem, Carathéodory's Theorem, Brion's Theorem, Minkowski's Convex Body Theorem, Bipolar Theorem, Pick's Formula, Diet Problem, Applying Lemma, Radon's Theorem, Sketch of Proof, Minkowski's Theorem, Neumann Theorem, Dvoretzky's Theorem, Recognizing Points New!
Books on Related Topics | Concordance | Text Stats Browse Sample Pages:
Front Cover | Table of Contents | First Pages | Index | Back Cover | Surprise Me!
Citations (learn more)
This book cites 29
books:
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8
books
cite this book:
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- Convex Polytopes (London Mathematical Society Lecture Note Series) by P. McMullen in Back Matter
- Convex Bodies: The Brunn-Minkowski Theory (Encyclopedia of Mathematics and its Applications) by Rolf Schneider in Back Matter
- Introduction to Toric Varieties. (AM-131) by William Fulton in Back Matter
- Geometry of Numbers, Second Edition (North-Holland Mathematical Library) by C.G. Lekkerkerker in Back Matter
- Introduction to Geometric Probability (Lezioni Lincee) by Daniel A. Klain in Back Matter
See all 29 books this book cites
- Positivity in Algebraic Geometry I: Classical Setting: Line Bundles and Linear Series (Ergebnisse der Mathematik Und Ihrer Grenzgebiete) by R.K. Lazarsfeld in Back Matter
- Experimentation in Mathematics: Computational Paths to Discovery by Jonathan M. Borwein in Back Matter
- Convex Optimization & Euclidean Distance Geometry by Jon Dattorro in Back Matter
- Positivity in Algebraic Geometry I: Classical Setting: Line Bundles and Linear Series (Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge A Series of Modern Surveys in Mathematics) by R.K. Lazarsfeld in Back Matter
- Robust Optimization-Directed Design (Nonconvex Optimization and Its Applications) by Andrew J. Kurdila in Back Matter
See all 8 books citing this book
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