##
**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.

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) |