This book provides an excellent and exhaustive description of the theory of asymptotic geometric analysis. Although this important field moves between (finite-dimensional) convex geometry and (infinite-dimensional) functional analysis, both the techniques and the subject matter in asymptotic geometric analysis are completely different from those in geometry and functional analysis. Moreover, the intuition breaks this topic. Thus, a special characteristic of this theory is that the usual “isometric” geometric problems are replaced by “isomorphic” questions, which make sense only from an asymptotic point of view. We can find a remarkable example of this in the isoperimetric type problems: isomorphic isoperimetric inequalities led to the discovery of the so-called concentration of measure phenomenon which plays a crucial role in this theory. This phenomenon is also “guilty” for the above mentioned breaking of intuition.
Another illustrative example showing the difference in behavior between the finite-dimensional and the high-dimensional cases is given by the following famous question, known as the hyperplane conjecture: does there exist a dimension independent constant \(c>0\) such that any convex body with volume one has a hyperplane section of measure greater than or equal to \(c\)? Whereas in a fixed dimension \(n\) a typical compactness argument provides a positive answer, when the question is posed for any (high) dimension, there is no solution nowadays. It is currently one of the most challenging and beautiful open questions in this field, and the book under review deeply deals with this issue in Chapter 10.
The contents of the first two chapters provide the background material from convex geometry. Indeed, convexity is always strongly behind the development of asymptotic geometric analysis, and thus it has a significant presence throughout the book. In particular, and among other questions, Chapter 1 is devoted to studying the Brunn-Minkowski inequality: several enlightening proofs and applications of it are collected, such as the Rogers-Shephard, Blaschke-Santaló and isoperimetric inequalities, Borell’s lemma, etc.; the functional form of the Brunn-Minkowski inequality, the so-called Prékopa-Leindler inequality, is also considered. In fact, the functional analytic point of view is always present in the book, because the existing interplay between convex bodies and functions is particularly fruitful. In Chapter 2 classical positions of convex bodies are investigated, namely, John position, minimal surface area position and minimal mean width position. As a significant application of John position, the reverse isoperimetric inequality is proved, for which Ball’s normalized form for the Brascamp-Lieb inequality, as well as its reverse form by Barthe, are studied. The book even ends with two appendices devoted to considering basic facts from elementary convexity, convex functions and mixed volumes theory (Steiner symmetrization, Alexandrov-Fenchel inequality, Steiner’s and Kubota’s formulae, etc.).
Chapter 3 deals with what may be described as one of the cores of this theory: the concentration of measure phenomenon, which was put forward in the early 1970s by V. Milman. This phenomenon became soon very valuable, not only in asymptotic geometric analysis, but also in other fields like probability theory or asymptotic combinatorics. In this book, the authors present several approaches to this problem, starting with the enlightening examples of isoperimetric inequalities in the sphere, the Gauss space and the discrete cube. In these cases the approach is made through the extremal sets, and then the concentration inequalities arise as a computational consequence. This approach however does not always work, since the extremal sets often cannot be determined, and thus the authors investigate different ways to get concentration inequalities as, for instance, using some compact groups and their homogeneous spaces or conditional expectation and martingales.
In Chapter 4 the covering numbers and the entropy numbers are introduced, and many properties, as well as duality relations between them and useful estimates in terms of other parameters of the sets, are studied.
The Dvoretzky-Milman theorem states that for any normed space \(X=\bigl({\mathbb R}^n,\|\cdot\|\bigr)\) and every \(0<\varepsilon<1\), there exist an integer \(k\geq c\varepsilon^2\log n\) and a \(k\)-dimensional subspace \(F\) of \(X\) such that \(d(F,\ell^k_2)\leq 1+\varepsilon\); or geometrically speaking: every high-dimensional centrally symmetric convex body has a central section of high dimension which is almost an ellipsoid (in the sense of small Banach-Mazur distance). Chapter 5 is devoted to deeply studying this important theorem as well as the main developments around it (as the volume ratio theorem). Milman’s proof of this result smartly uses the concentration measure phenomenon for the Euclidean sphere, and provides an asymptotic formula for such a \(k\), given in terms of the dimension \(n\), and the average and the Lipschitz constant of the norm, which is sharp in full generality.
In Chapter 6 the authors deal with estimates for the product of the mean width of a body and the mean width of its polar. They prove that it can be bounded from above by the \(K\)-convexity constant, which in turn can be bounded using Pisier’s inequality, one of the fundamental facts in the local theory of normed spaces. Consequences of this estimate are also studied, as for instance the “reverse Urysohn inequality”. Replacing the Gaussian measure by the uniform measure on the discrete cube, another bound is obtained for the above product, the so-called Rademacher constant, which is proved to be equivalent to the \(K\)-convexity constant.
Results on proportional subspaces and quotients of an \(n\)-dimensional normed space are considered in Chapter 7, this is, those having dimension \(\lambda n\), for \(\lambda\in(0,1)\). Milman’s \(M^*\)-estimate was the first step in this direction: geometrically speaking it says that the diameter of a random proportional section of a centrally symmetric convex body is controlled by its mean width. Several proofs of this theorem are provided, as well as improvements and deep consequences of it. Among them, the Milman quotient of subspace theorem is stressed, playing also a crucial role in its proof Pisier’s inequality, which was studied in the previous chapter.
Chapter 8 is devoted to one of the most significant results in asymptotic geometric analysis: the existence, for any convex body, of an \(M\)-position, this is, an ellipsoid with the same volume that can replace the body in many computations (up to universal constants). This fact, discovered by Milman in 1986, became a strong tool in this field, and has important consequences such as the reverse Brunn-Minkowski inequality. The chapter starts describing a proof of the reverse Santaló inequality, which is known as the Bourgain-Milman inequality, and continues with an alternative proof of this result based on the Milman isomorphic symmetrization technique. Then, the authors focus on the already mentioned deep result about the existence of an \(M\)-position and its valuable consequence: the Milman reverse Brunn-Minkowski inequality. Further applications and Pisier’s approach to these results end the chapter.
Different methods in the study of random processes are strongly connected with asymptotic geometric analysis. Thus, the main aim of Chapter 9 is to introduce a Gaussian approach to several results previously studied, as for instance, the Dvoretzky-Milman theorem or the \(M^*\)-estimate: the proofs will be based on Gordon’s comparison principles for Gaussian processes.
The last chapter of the book, Chapter 10, is a collection of recent discoveries and significant open problems on the distribution of volume in high-dimensional convex bodies. This discussion is made in the natural framework of convex bodies in isotropic position. Certainly, one of the major unsolved problems in this context is the slicing problem (or hyperplane conjecture), already mentioned at the beginning of this review, which has an equivalent formulation in terms of the isotropic constant: \(L_K\) is bounded from above by a constant independent of the dimension. The chapter starts discussing this important question, and moves on to the more general setting of finite log-concave measures. Then the authors present a number of important results in connection with these issues, namely, the best known upper bounds for the isotropic constant (Bourgain and Klartag estimates), the solution to an isomorphic slicing problem by Klartag, or Paouris’ deviation inequality (along with the theory of \(L_q\)-centroid bodies).
All chapters are enriched with a final section, a collection of “Notes and remarks”, where the main references regarding the content of the chapter can be found, as well as many applications and problems related to its subject. The list of references is large and exhaustive, consisting of more that 600 items.
This book (and its second volume) has become the essential reference and main source for the theory of asymptotic geometric analysis.

Reviewer: Maria A. Hernández Cifre (Murcia)