## The volume of convex bodies and Banach space geometry.(English)Zbl 0698.46008

Cambridge Tracts in Mathematics, 94. Cambridge etc.: Cambridge University Press. xv, 250 p. £30.00; \$ 49.50 (1989).
The aim of the book is to give a self-contained presentation of results recently obtained, relating to the volume of convex bodies in $${\mathbb{R}}^ n$$ and the geometry of the corresponding finite-dimensional normed spaces.
The first part (8 out of the 15 chapters) elaborates proofs of the following three fundamental results:
(I) The “quotient of subspace theorem” due to V. D. Milman, Proc. Am. Math. Soc. 94, 445-449 (1985; Zbl 0581.46014): For each $$0<\delta <1$$ there is a constant $$C=C(\delta)$$ such that every n-dimensional normed space E admits subspaces $$E_ 1$$, $$E_ 2$$ $$(E_ 2\subset E_ 1)$$ for which $$F=E_ 1/E_ 2$$ has dim $$F\geq \delta n$$ and is C-isomorphic to an Euclidean space $$E'=(\ell^ p_ 2)$$, i.e., $$\inf \{\| T\| \cdot \| T^{-1}\| \}\leq C$$, where T sums over all isomorphisms $$F\to E'.$$
(II) The “inverse Santaló inequality” due to J. Bourgain and V. D. Milman, Invent. Math. 88, 319-340 (1987; Zbl 0617.52006): There are positive constants $$\alpha$$, $$\beta$$ (independent of n) such that for all balls $$B\subset {\mathbb{R}}^ n$$ (i.e., compact convex symmetric bodies admitting the origin as an interior point) we have $\alpha /n\leq (vol B\cdot vol B\circ)^{1/n}\leq \beta /n,$ where $$B\circ$$ denotes to polar of B (the upper bound goes back to L. Santaló).
(III) The “inverse Brunn-Minkowski inequality” due to V. Milman, C. R. Acad. Sci. Paris 302, Sér. 1, 25-28 (1986; Zbl 0604.52003)]: Two balls $$B_ 1$$, $$B_ 2$$ in $${\mathbb{R}}^ n$$ can always be transformed by a volume preserving linear isomorphism into balls $$B^{\sim}_ 1$$, $$B^{\sim}_ 2$$ which satisfy $[vol(B^{\sim}_ 1+B^{\sim}_ 2)]^{1/n}\leq C[(vol B^{\sim}_ 1)^{1/n}+(vol B^{\sim}_ 2)]^{1/n},$ where C is a numerical constant independent of n. Moreover, the polars of $$B_ 1$$, $$B_ 2$$ and all their multiples also satisfy a similar inequality.
Chapter 9 is studying the “volume numbers” of an operator T: $$X\to \ell_ 2$$ (X a Banach space) defined by $v_ n(T)=\sup [(vol P\overline{TB_ X})/V_ n]^{1/n},$ where B is the unit ball of X, P runs over the set of orthogonal projections of rank n on $$\ell_ 2$$ and $$V_ n$$ denotes the volume of the canonical unit ball. These numbers are compared with the “entropy numbers” of T and shown that they are “almost equivalent”. The main reference here is the paper of V. Milman and G. Pisier, Ann. Probab. 15, 292-304 (1987; Zbl 0636.60036).
The following chapters deal with special kind of Banach spaces X, in particular, the so-called type p (1$$\leq p\leq 2)$$ and cotype q $$(2\leq q<\infty)$$ spaces. These are respectively characterized by the existence of constants C (depending on the space X and on p or q) such that $\| \sum^{n}_{1}g_ ix_ i\|_{L_ 2(X)}\leq C(\sum^{n}_{1}\| x^ p_ i\|)^{1/p}\quad or\quad (\sum^{n}_{1}\| x_ i\|^ q)^{1/q}\leq C\| \sum^{n}_{1}g_ ix_ i\|_{L^ 2(X)},$ for any finite set $$\{x_ i\}^ n_ 1\subset X$$ and a corresponding set $$\{g_ i\}^ n_ 1$$ of independent identically distributed Gaussian random variables with the standard normal distribution. In case $$p=2$$ or $$q=2$$ these relations can be transformed in a from which suggests the definition of weak cotype 2 and weak type 2 spaces. If X is of both these types, it is called a weak Hilbert space. Chapters 13-15 are given a study of these spaces. In particular it is shown that weak Hilbert spaces are reflexive (W. B. Johnson) and enjoy the uniform approximation property [G. Pisier, Proc. London Math. Soc. 56, 547-579 (1988; Zbl 0666.46009)].
The book ends with remarks on some other open problems which have recently received great deal of attention.
Reviewer: B.Sz.-Nagy

### MSC:

 46B20 Geometry and structure of normed linear spaces 52A20 Convex sets in $$n$$ dimensions (including convex hypersurfaces) 46-02 Research exposition (monographs, survey articles) pertaining to functional analysis 52-02 Research exposition (monographs, survey articles) pertaining to convex and discrete geometry 52C07 Lattices and convex bodies in $$n$$ dimensions (aspects of discrete geometry)