x Preface the general view of infinite-dimensional linear programming (Chapter IV) from him. This approach makes the exposition rather simple and elegant. It makes it possible to deduce a variety of strong duality results from a single simple theorem (Theorem IV.7.2). My interest in quadratic convexity (Section II.14) and other “hidden convexity” results (Section III.7) was inspired by him. He also encouraged my preoccupation with lattice points (Chapter VIII) and various peculiar polytopes (Sections II.5–7). On various stages of the project I received encouragement from A. Bj¨ orner, L. Billera, R. Pollack, V. Klee, J.E. Goodman, G. Kalai, A. Frieze, L. Lov´ asz, W.T. Gowers and I. B´ ar´any. I am grateful to my colleagues in the Department of Mathematics at the Univer- sity of Michigan in Ann Arbor, especially to P. Hanlon, B.A. Taylor, J. Stembridge and S. Fomin with whose blessings I promoted convexity within the Michigan com- binatorics curriculum. I thank the students who took Math 669 convexity classes in 1994–2001. Special thanks to G. Blekherman who contributed some of his in- teresting results on the metric structure of the set of non-negative multivariate polynomials (Problems 8 and 9 of Section V.2.4). Since the draft of this book was posted on the web, I received very useful and detailed comments from R. Connelly, N. Ivanov, J. Lawrence, L. Lov´ asz, G. Ziegler and A.M. Vershik. I am particularly grateful to J. Lawrence who suggested a number of essential improvements, among them are the greater generality of the “polarity as a valuation” theorem (Theorem IV.1.5), a simplified proof of the Euler-Poincar´ e Formula (Corollary VI.3.2) and an elegant proof of Gram’s relations (Problem 1 of Section VIII.4.4) and many mathematical, stylistic and bibliograph- ical corrections. I thank A. Yong for reading the whole manuscript carefully and suggesting numerous mathematical and stylistic corrections. I thank M. Wendt for catching a mistake and alerting me by e-mail. I thank S. Gelfand (AMS) for insisting over a number of years that I write the book and for believing that I was able to finish it. I am grateful to the National Science Foundation for its support. Ann Arbor, 2002 Alexander Barvinok

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