Preface

Asymptotic convex geometry may be described as the study of convex bodies

from a geometric and analytic point of view, with an emphasis on the dependence

of various parameters on the dimension. This theory stands at the intersection

of classical convex geometry and the local theory of Banach spaces, but it is also

closely linked to many other fields, such as probability theory, partial differential

equations, Riemannian geometry, harmonic analysis and combinatorics. The aim

of this book is to introduce a number of basic questions regarding the distribution

of volume in high-dimensional convex bodies and to provide an up to date account

of the progress that has been made in the last fifteen years. It is now understood

that the convexity assumption forces most of the volume of a body to be concen-

trated in some canonical way and the main question is whether, under some natural

normalization, the answer to many fundamental questions should be independent

of the dimension.

One such normalization, that in many cases facilitates the study of volume

distribution, is the isotropic position. A convex body K in

Rn

is called isotropic if

it has volume 1, barycenter at the origin, and its inertia matrix is a multiple of the

identity: there exists a constant LK 0 such that

K

x, θ

2dx

=

LK2

for every θ in the Euclidean unit sphere

Sn−1.

It is easily verified that the aﬃne

class of any convex body K contains a unique, up to orthogonal transformations,

isotropic convex body; this is the isotropic position of K. A first example of the

role and significance of the isotropic position may be given through the hyperplane

conjecture (or slicing problem), which is one of the main problems in the asymptotic

theory of convex bodies, and asks if there exists an absolute constant c 0 such

that maxθ∈Sn−1 |K ∩

θ⊥|

c for every convex body K of volume 1 in

Rn

that

has barycenter at the origin. This question was posed by Bourgain [99], who was

interested in finding Lp-bounds for maximal operators defined in terms of arbitrary

convex bodies. It is not so hard to check that answering his question aﬃrmatively

is equivalent to the following statement:

Isotropic constant conjecture. There exists an absolute constant C 0 such

that

Ln := max{LK : K is isotropic in

Rn}

C.

This problem became well-known due to an article of V. Milman and Pajor

which remains a classical reference on the subject. Around the same time, K. Ball

showed in his PhD Thesis that the notion of the isotropic constant and the conjec-

ture can be reformulated in the language of logarithmically-concave (or log-concave

ix