CHAPTER 1 Background from asymptotic convex geometry In this introductory chapter we survey the prerequisites from the theory of convex bodies and the asymptotic theory of finite dimensional normed spaces. Short proofs are provided for the most important results that are used in the sequel. Basic references on convex geometry are the monographs by Schneider [463] and Gruber [237]. The asymptotic theory of finite dimensional normed spaces is presented in the books by V. Milman and Schechtman [387], Pisier [430] and Tomczak- Jaegermann [493]. The books of Rockafellar [442], Bogachev [92], Brezis [121] and Feller [169], [170] are very useful sources of information on facts from convex analysis, functional analysis and probability theory that are being used throughout this book. The first four sections of this chapter contain background material from classical convexity: the Brunn-Minkowski inequality and its functional forms, mixed volumes and classical geometric inequalities. Section 1.5 introduces three classical positions of convex bodies: John’s posi- tion, the minimal mean width position and the minimal surface area position. All of them arise as solutions of extremal problems and can be characterized as satis- fying an isotropic condition with respect to an appropriate measure. This relates them to the Brascamp-Lieb inequality and its reverse. In Section 1.6 we discuss Barthe’s proof of these inequalities and their applications to geometric problems an example is K. Ball’s sharp reverse isoperimetric inequality. Section 1.7 introduces the concept of measure concentration and the main ex- amples of metric probability spaces that will be used in this book: the sphere, the Gauss space and the discrete cube. The next two sections survey basic probabilistic tools that we will use: covering numbers and basic inequalities for them, Gaussian and sub-Gaussian processes and bounds for the expectation of their supremum. The last sections of the chapter give a brief synopsis of the major results of asymptotic convex geometry: Dvoretzky type theorems, the notion of volume ratio and Kashin’s theorem, the -position and Pisier’s inequality on the Rademacher projection, the MM -estimate, Milman’s low M -estimate and the quotient of subspace theorem. Finally, we present the reverse Santal´ o inequality and the reverse Brunn-Minkowski inequality during this discussion M-ellipsoids and their basic properties are also introduced. 1.1. Convex bodies We work in Rn, which is equipped with a Euclidean structure ·, ·. We denote by · 2 the corresponding Euclidean norm, and write B2 n for the Euclidean unit ball and Sn−1 for the unit sphere. Volume is denoted by | · |. We write ωn for the volume of B2 n and σ for the rotationally invariant probability measure on Sn−1. 1
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