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Convex Bodies: The Brunn-Minkowski Theory

Rolf Schneider
Cambridge University Press
Publication Date: 
Number of Pages: 
Encyclopedia of Mathematics and Its Applications 151
[Reviewed by
Michael Berg
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When I was a UCLA freshman I learned about Minkowski’s theorem about convex sets and lattice points in a number theory course taught by the redoubtable Ernst Straus (1922–1983). The result in question is that if a convex set in a Euclidean n-space has a volume of at least \(2^n\) and is symmetric about the origin, then that set must contain lattice points other than the origin. It’s a gorgeous result, and Minkowski’s proof is truly elegant; see, e.g., the wonderful exposition in Cassels’s An Introduction to the Geometry of Numbers. Prior to this I had only heard about Minkowski in connection with Einstein’s work on general relativity although he was featured as a major player in what was then (and remains) my favorite mathematical biography, Constance Reid’s Hilbert. What quickly dawned on me was that, even among the titans, Minkowski was an unusual mathematician in many ways, specifically in light of his breadth and his penchant for providing beautiful geometric proofs of analytic or even arithmetical facts. This is borne out in any number of ways; for example, consider his famous inequality about \(L^p\)-spaces, whose proof, while not strictly speaking geometrical, does involve convexity, and the subject of the book under review, the Brunn-Minkowski theorem, where convexity is again featured. By the way, according to Wikipedia — a dicey source, of course — it was Lyusternik who extended the result to non-convex sets in 1935.

Well, what then does the Brunn-Minkowski theorem assert? In the book under review, which is nothing if not encyclopedic and thorough (it weighs in at over 700 pages after all), the statement comes on p. 369, after a lot of preliminary work has been done. Here is the more accessible (and compact) statement given on p. 371: if \(K^0\) and \(K^1\) are convex bodies in \(\mathbf{R}^n\) and \(V_n\) denotes n-dimensional volume, then for all non-negative reals s, t one gets that \(V_n(s K^0 + tK^1)^{1/n} \geq sV_n(K^0)^{1/n} + tV_n(K^1)^{1/n}\). (It is irresistible, isn’t it, to note how much this inequality resembles Minkowski’s \(L^p\) inequality, modulo the inequality facing the other way — the wonders of convexity, I guess…) So here, then, is the star of the play. But, again, he doesn’t appear on stage till about half-way through. What’s happening before that?

The answer is that the author, Rolf Schneider, does true justice to the call of crafting a volume of the Encyclopedia of Mathematics and Its Applications and develops the story meticulously and exceedingly thoroughly. The subjects of convexity, Minkowski addition (of convex bodies), and volume are dealt with very, very carefully ere we get to Brunn-Minkowski proper, with everything motivated and defined, and theorems carefully stated and proven. And this sets the tone for all that ensues, both the treatment of the Brunn-Minkowski theorem itself and the dividends one reaps from it (and its equivalents). For example, Schneider presents us with discussions of the role played by area measures and curvatures as well as themes involving area construction. Finally it is important to note that an entire later chapter is devoted to “extensions and analogues of the Brunn-Minkowski theory,” including the \(L^p\) theory and a dual theory. (And, guess what, in this latter context we find discussions of both Hölder’s inequality and, yes, Minkowski’s inequality, all in the \(L^p\) setting — Indeed, the wonders of convexity.)

What else need I say? It’s a titanic effort, and a successful one. Schneider peppers the book with historical notes and references to original works, he writes wonderfully informative prelude sections to his chapters, he has added indices on notation, authors, and subjects, and he provides the reader with a daunting bibliography of no fewer than 2083 (yes, it’s a prime) entries, ranging from Abardia to Zymonopoulou. This is truly an encyclopedic volume. It is also fine scholarship and any one dealing with these themes needs to have this book in his possession.

Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles, CA.

Preface to the second edition
Preface to the first edition
General hints to the literature
Conventions and notation
1. Basic convexity
2. Boundary structure
3. Minkowski addition
4. Support measures and intrinsic volumes
5. Mixed volumes and related concepts
6. Valuations on convex bodies
7. Inequalities for mixed volumes
8. Determination by area measures and curvatures
9. Extensions and analogues of the Brunn–Minkowski theory
10. Affine constructions and inequalities
Appendix. Spherical harmonics
Notation index
Author index
Subject index.