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 2 dx = 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
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