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  and Gruber
. The asymptotic theory of finite dimensional normed spaces is presented
in the books by V. Milman and Schechtman , Pisier  and Tomczak-
The books of Rockafellar , Bogachev , Brezis  and Feller ,
 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
which is equipped with a Euclidean structure ·, ·. We denote
the corresponding Euclidean norm, and write B2
for the Euclidean unit
for the unit sphere. Volume is denoted by | · |. We write ωn for the
volume of B2
and σ for the rotationally invariant probability measure on