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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)
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