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
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
x, θ
for every θ in the Euclidean unit sphere
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
c for every convex body K of volume 1 in
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
Ln := max{LK : K is isotropic in
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
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