In the last decade, interest in the problem of bounds for volumes of convex bodies was renewed mainly because of its applications to Banach Space Geometry and related topics. At the end of the 80's sharp bounds for volume radius of convex polytopes with given distance between antipodal faces were found independently by several authors: Carl and Pajor [1], Bourgain, Lindenstrauss and Milman [2], Gluskin [3]. Closely related results were obtained by Vaaler [4], Dilworth and Szarek [5] and Baniny and Furedi [6]. See also Ball and Pajor [7] where, following Kashin's conjecture, the problem was considered as a limiting case of a series of Vaaler-type results. Moreover in [3] it was observed that the volume radius of a unit cube has a certain stability property with respect to cutting the cube by a sequence of bands (see Proposition 1 below for the exact formulation). Some of Kashin's ideas enabled us to use this property for an alternative proof of Spencer's theorem [8] on a lacunary analogue of the RudinShapiro polynomials (see [3]). Later Kashin [9] used the same approach for finite dimensional analogues of Menshov's correction theorem.

1. Introduction and Statement of Main Results

Let K = (K1, K2 , … , Ks) be an s-tuple of compact convex subsets of ℝn . For any continuous function F : ℝn → ℂ, consider the function

This defines an operator M K, which we will call a Minkowski operator. Denote by A (ℂ n) the Frechet space of entire functions in n variables with the usual topology of the uniform convergence on compact sets in ℂ n, and Cr (ℝn) the Frechet space of r times differentiable functions on ℝn with the topology of the uniform convergence on compact sets in ℝn of all partial derivatives up to the order r (1 ≤ r ≤ ∞). The main results of this work are Theorems 1 and 3 below.

1. Introduction

It is an open problem whether there exists an infinite-dimensional Banach space X such that every bounded linear operator from X to itself is of the form ƛI + K, where).ƛ. is a scalar, I is the identity on X and K is a compact operator. The strongest property of a similar nature that has been obtained is that a space may be hereditarily indecomposable (see [GM] for this definition and several others throughout the paper), which implies [GM] that every operator on it is of the form

ƛI + S, where S is strictly singular, and even [Fl] that every operator from a subspace into the space is a strictly singular perturbation of a multiple of the inclusion map. (These results assume complex scalars but several examples are known where the conclusion holds with real scalars.) In this note, we show that the first hereditarily indecomposable space to be discovered [GMJ, which we shall call X, has a subspace Y such that there is a non-compact strictly singular operator from Y into X. Therefore this operator is not a compact perturbation of a multiple of the inclusion map. Since all we are doing is showing that one particular space does not give an example of a stronger property than that required by the problem, the existence of this note needs some justification, which we shall now provide.

First, if one is trying to solve the problem with an example, then a natural line of attack is to try to construct a hereditarily indecomposable space such that every strictly singular operator is compact.

Let V be a finite-dimensional vector space over ℝ and let V* denote the dual vector space. A symmetric convex body or (Banach) ball is a compact convex set with nonempty interior which is invariant under under x ⟼ -x. We define Ko ⊂ V*, the dual of a ball K ⊂ V, by A ball K is the unit ball of a unique Banach norm ||·|| K defined by A ball K is an ellipsoid if ||·|| K is an inner-product norm. Note that all ellipsoids are equivalent under the action of GL(V).

If V is not given with a volume form, then a volume such as Vol K for K ⊂ V is undefined. However, some expressions such as (Vol K)(Vol Ko) or (Vol K) / (Vol K’) for K, K’ ⊂ V are well-defined, because they are independent of the choice of a volume form on V, or equivalently because they are invariant under GL(V) if a volume form is chosen. An r -dimensional ball K is r-semiround [8] if it contains an ellipsoid E such that It is r-round if it contains an ellipsoid E such that K ⊆ r E. Santal6's inequality states that if K is an n-dimensional ball and E is an n-dimensional ellipsoid.

1. Introduction

Denote by VOli (.) the i-dimensional Lebesgue measure, and denote by Gi,n the Grassmann manifold of i-dimensional subspaces ofℝ n. The generalized Busemann-Petty problem asks: G BP. If K and L are origin-symmetric convex bodies in ℝ n, is there the implication

The case of i = 1 is trivially true. The hyperplane case i = n - 1 is well-known as the Busemann-Petty problem (see [BP] and [Bu]). Many authors contributed to the solution of the Busemann-Petty problem (see [Ba] [Bo] [Gl] [Gia] [Gie] [GR] [Ha] [Lu] [LR] [Pal [ZI]). The problem has a negative answer when n > 4 (see [Gl], [Pal and [Z2]) , and it has a positive answer when n = 3,4 (see [G2] and [Z4]).

1. Introduction

The classical spaces ℓp and Lp are the best known and in many ways most fundamental examples of Banach spaces. In view of their interesting properties it is natural to ask whether the role of the function tP in these spaces can be played by other more general functions. This question was answered by Orlicz, who defined a certain class of functions, now known as Orlicz functions, and associated with each one a sequence space and a function space, now called an Orlicz sequence space and Orlicz function space. The Orlicz spaces are generally regarded as the correct and most natural spaces to associate with given Orlicz functions.

One of the aims of this paper is to cast doubt on that view, at least in its isometric interpretation. We shall do this by discussing a different generalization which arises geometrically and has two desirable isometric properties lacked by Orlicz spaces. First, the dual of one of our spaces is isometric to another such space. Second, complex interpolation between two of our spaces yields a third in a natural way. Irritatingly, we have not managed to establish whether our new spaces are isomorphic to Orlicz spaces, in which case they are a useful renorming of them, or whether they are completely different. Our route to the new generalization starts with an unusual (perhaps even eccentric) problem which will be described below, and which relates more to the polytope approximations of the title.

1. Definition of Curvature Let(Ω, (μ) be a probability space and L a self-adjoint negative but not necessarily bounded operator on L2 (μ) given by

where K is a non negative symmetric kernel. Obviously L remains unchanged if we change K on the diagonal. By P t = etL we denote the continuous contraction semigroup on L2(μ) with generator L. We will assume that Pt is ergodic and that there exists an algebra A⊆∩n dom Ln of bounded functions which is a form core of L. Then the Beurling-Deny condition implies that Pt is a symmetric Markov semigroup, i.e., Pt preserves positivity and extends to a continuous contraction semigroup on Lp(μ) for all 1 ≤p < ∞. We will also assume that A is stable under Pt.

The purpose of this note is to present a localization technique for the sphere sn on an example of the Gromov-Milman theorem [Gr-M] about the concentration phenomenon on uniformly convex spheres. This result was obtained in [Gr-M] in a some more general setting. Our approach follows the same general reasoning, but is simpler and more direct than the original approach. We also outline Rohlin's theory of measurable partitions, which is used in the proof. Note that the terminology of “localization” was introduced for lRn by L. Lovasz and M. Simonovits [L-Sl, L-S2]. [Gr-M] did not use such terminology and also did not put the scheme of localization explicitly.

NOTE. K. Ball has informed us recently that he, jointly with R. Villa, found an extremely short proof of the Gromov-Milman theorem for uniformly convex sphere as an application of the Prekopa-Leindler inequality.

The first edition of this book was published in 1980 in the LMS Lecture Note Series, and a Russian translation by V.V. Peller and A.G. Tumarkin, made under the direction of V.P. Havin, the editor, appeared in 1984.

Both versions of the book are now out of print, and for the past couple of years people have been asking me how they might procure a copy of it. The Cambridge University Press has therefore decided to put out a second edition, and I am grateful to Dr. David Tranah, the Press' senior mathematics editor, for his having arranged to issue it in somewhat improved typographical format as a Cambridge Tract.

In preparing the first edition I had tried to make the exposition as accessible as I could by concentrating on what I thought were the main ideas in the subject rather than on including as many results as possible. The readers I had in mind were those with some training in analysis who were trying to gain a secure foothold in the theory of Hp spaces, whether with the aim of eventually doing serious work in that subject or for the purpose of understanding its applications in other areas (e.g. in operator theory – some of the material is now even used in electrical engineering). I have been guided by the same concern while working on the second edition and have for that reason tried to preserve the book's original character.