1. Preliminaries

1.1. Notation. We follow standard Banach-space theory notation, as outlined in [LTz]. Throughout this note X will be an infinite dimensional Banach-space with a shrinking basis

The notation [X]n will stand for the head subspace (span and for the tail subspace (the closure of span Pn and P>n are the coordinate-orthogonal projections on these subspaces respectively.

A vector is called a block if it has finite support, that is, if it is a finite linear combination of elements of the basis. The blocks v and ware said to be consecutive (υ < w) if the support of υ (the set of elements of the basis that form υ as a linear combination) ends before the support of w begins. is the collection of all n-tuples of consecutive normalised blocks; thus means normalised finite support vectors.

1.2. An intuitive introduction to asymptotic structure. The language of asymptotic structure has been introduced to study the essentially infinite dimensional structure of Banach-spaces, and to help bridge between finite dimensional and infinite dimensional theories. This approach does generalise spreading models, but takes an essentially different view. Formally introduced in [MMiT], it has already been studied, extended and applied in [KOS], [OTW], [T], [W1] and [W2]; it is closely related to the new surge of results in infinite dimensional Banach-space theory, and especially to [G] and [MiT].

1. Introduction and Notation

Suppose that C is a convex body in ℝn such that 0 is an interior point of C, then the mean width ω( C) is defined by where Cn is a constant depending only on the dimension, σ a the normalized Haar measure on the sphere sn-1 and ‘ϒn the n-dimensional standard gaussian measure. Denoting by C* the polar of C with respect to 0 and by ||.|| c the gauge of C, we obtain the well known formula

The euclidean ball B n 2 is the Löwner ellipsoid of C if and only if B n 2 is the John ellipsoid of C* i.e., the ellipsoid of maximal volume contained in C*. Hence, in order to prove that the regular simplex has minimal mean width, it is enough to prove that for all convex bodies K whose John ellipsoid is the euclidean ball, we necessarily have 𝓁(K) ≥ 𝓁 (T), i.e., the 𝓁 -norm of K is bounded from below by the 𝓁 -norm of the regular simplex T.

Recently, a number of results of the Local Theory have been extended to the quasi-normed spaces. There are several works [KalI, Ka12, D, GL, KT, GK, BBP1, BBP2, M2] where such results as Dvoretzky-Rogers lemma [DvR], Dvoretzky theorem [Dv1, Dv2], Milman's subspace-quotient theorem [M1], Krivine's theorem [Kr], Pisier's abstract version of Grotendick's theorem [P1, P2], Gluskin's theorem on Minkowski compact urn [G], Milman's reverse Brunn-Minkowski inequality [M3], and Milman's isomorphic regularization theorem [M4] are extended to quasi-normed spaces after they were established for normed spaces. It is somewhat surprising since the first proofs of these facts substantially used convexity and duality.

In [AM2] D. Amir and V. D. Milman proved the local version of Krivine's theorem (see also [Gow], [MS]). They studied quantitative estimates appearing in this theorem. We extend their result to the q- and quasi-normed spaces. Recall that a quasi-norm on a real vector space X is a map ||·|| : X → ℝ+ satisfying these conditions: Note that I-norm is the usual norm. It is obvious that every q-norm is a quasi-norm with C = 2 1 / q-1 . However, not every quasi-norm is q-norm for some q. Moreover, it is even not necessary continuous. It can be shown by the following simple example.

The first example of dichotomy related to the topic discussed in this note is the classical combinatorial result of Ramsey: for every set A of pairs of integers, there exists an infinite subset M of ℕ such that, either every pair ﹛m1, m2﹜ from M is in A, or no pair from M is in the set A. There exist various generalizations to “infinite Ramsey theorems” for sets of finite or infinite sequences of integers, beginning with the result of Nash-Williams [NW]: for any set A of finite increasing sequences of integers, there exists an infinite subset M of N such that either no finite sequence from M is in A, or every infinite increasing sequence from M has some initial segment in A (although it does not look so at the first glance, notice that the result is symmetric in A and AC, the complementary set of A; for further developments, see also [GPl, [E]). The first naive attempt to generalize this result to a vector space setting would be to ask the following question: given a normed space X with a basis, and a set A of finite sequences of blocks in X (i.e., finite sequences of vectors (x1, … , xk) where x1, … , xk ∈ X are successive linear combinations from the given basis), does there exist a vector subspace Y of X spanned by a block basis, such that either every infinite sequence of blocks from Y has some initial segment in A, or no finite sequence of blocks from Y belongs to A, up to some obviously necessary perturbation involving the norm of X.

Recall that a p-norm on real vector space X is a map 11·11 : X -+ ℝ+ satisfying these conditions:

• (1) ||xll > 0 for all x ≠ 0.

• (2) ||tx|| = |t|||x|| for all t ∈ ℝ and x ∈ X.

• (3) ||x + y||P ≤ ||x||P + ||y||P for all x, y ∈X.

Note that the unit ball of p-normed space is a p-ball and, vice versa, the gauge of p-ball is a p-norm.

Recently, J. Bastero, J. Bernues, and A. Peiia [BBP] extended the reverse Brunn-Minkowski inequality, which was discovered by V. Milman [MJ, to the class of p-convex balls. They proved the following result.

1. Introduction The purpose of this paper is to prove various geometric inequalities in option pricing using familiar inequalities of the Brunn-Minkowski type in Gauss space. To begin with, recall that a European (American) call [putJ option is defined as the right to buy [sellJ one share of stock at a specified price on (or before) a specified date. The specified price is referred to as the exercise price and the terminal date of the contract is called the expiration date or maturity date. In fact, already the early paper [20J by Merton treats a variety of convexity properties of puts and calls, sometimes without any distributional assumptions on the underlying stock prices. Here, however, it will always be assumed that the price process X(t) = (X1 (t), …,Xm(t)), t ≥ 0, of the underlying risky assets X l, … , Xm is governed by a so called joint geometric Brownian motion. Furthermore, all options will be of European type and so, from now on, option will always mean option of European type.

1. Introduction and Notation

Let A be a precompact subset of a metric space (X, τ). An E-net of A is a subset A of X such that any point x of A can be approximated by a point y of A such that τ (X, y) < ϵ. The smallest cardinality of an E-net of A is called the covering number of A and is denoted by N(A, τ, ϵ). The metric entropy (shortly the entropy) is the function log N(A, τ, ). When X is ad-dimensional normed space equipped with the metric associated to its norm ||.||, we will denote by N(A, ||.||, ∈) the covering number of a subset A of X and by B(X) the unit ball of X. The metric entropy of a ball A = rB(X) of radius r is computed by volumic method (see [MS] or [P]): for ϵ ∈[0, r],

Our main result is the following extension of Dvoretzky's theorem (from the range 1 < k < c log n to c log n ≤ k < n), first announced in [MS2, Theorem 2]. As is remarked in that paper, except for the absolute constant involved the result is best possible. THEOREM. There exists a K > 0 such that, for every n and every log n : ≤ k < n, any n-dimensional normed space, X, contains a k-dimensional subspace, Y, Satisfying

Jesus Bastero pointed out to us that the proof of the theorem in [MS2] works only in the range k ≤ cn/ log n. Here we give a different proof which corrects this oversight. The main addition is a computation due to E. Gluskin (see the proof of the Theorem in [Gll] and the remark following the proof of Theorem 2 in [G12]). In the next lemma we single out what we need from Gluskin's argument and sketch Gluskin's proof.

During the last ten years the integral geometry of convex bodies has undergone a dramatic revitalisation, brought about by the introduction of methods, results and, most importantly, new viewpoints, from probability theory, harmonic analysis and the geometry of finite-dimensional normed spaces. The principal goal of this program was to bring together researchers from several different fields, Classical Convex Geometry, Geometric Functional Analysis, Computational Geometry and related areas of Harmonic Analysis. The main reason for doing so was that research in these areas has found considerable overlap in recent years. Several problems and classes of problems have been come upon independently from different directions, and techniques from some areas have been found important in others. This goal was achieved beyond even our most optimistic expectations.

As well as an introductory workshop, consisting of four lecture series with an educational format, the program included one full-scale research workshop and two concentrations of visitors, in addition to the regular activity during the principal five months. About 190 mathematicians attended the program in some capacity or other and there were over 150 lectures and seminars during the period. These were of several types. There was a regular educational seminar, two or three times a week which enabled participants to become acquainted with material from other fields. Three or four lectures a week dealt with recent research by members, and there was a “young research seminar” (roughly once a week) which gave postdocs and students a chance to describe their work in an informal atmosphere.

If A is monotone, then μp(A) is clearly an increasing function of p. Considering A as a “property”, one observes in many cases a threshold phenomenon, in the sense that μp(A) jumps from near to near 1 in a short interval when n→∞. Well known examples of these phase transitions appear for instance in the theory of random graphs. A general understanding of such threshold effects has been pursued by various authors (see for instance Margulis [M] and Russo [RD. It turns out that this phenomenon occurs as soon as A depends little on each individual coordinate (Russo's zero-one law). A precise statement was given by Talagrand [T] in the form of the following inequality.

Define for i = 1, … , n where Ui(x) is obtained by replacement of the i-th coordinate Xi by 1 - Xi and leaving the other coordinates unchanged. The number μp(Ai) is the influence of the i-th coordinate (with respect to μp).

1. Introduction

Throughout this note K and L denote convex symmetric bodies in R.n. Our notation will be the standard notation that can be found, for example, in [2] and [4]. For 1 ≤ m ≤ n, Vm(K) denotes the m-th mixed volume of K (i.e., mixing m copies of K with n - m copies of the Euclidean ball Bn of radius one in R.n). Thus if m = n then Vn(K) = voln(K) and if m = 1 then V1(K) = ω(K) the mean width of K. For 0 < δ < 1 we define the m-th mixed 8-convolution body of the convex symmetric bodies K and L in R.n: DEFINITION. The m-th mixed δ-convolution body of K and L is defined to be the set It is a consequence of the Brunn-Minkowski inequality for mixed volumes that these bodies are convex.

Let Kt be the illumination bodies of K and Qn a polytope that contains K and has at most n (d-1)-dimensional faces.

1. Introduction

We investigate the approximation of a convex body K in Rd by a polytope. We measure the approximation by the symmetric difference metric. The symmetric difference metric between two convex bodies K and C is

We study in particular two questions: How well can a convex body K be approximated by a polytope Pn that is contained in K and has at most n vertices and how well can K be approximated by a polytope Qn that contains K and has at most n (d-1)-dimensional faces. Macbeath [Mac] showed that the Euclidean Ball Bd2 is an extremal case: The approximation for any other convex body is better. We have for the Euclidean ballprovided that n ≥ (C3 d)(d-1)/2. The right hand inequality was first established by Bronshtein and Ivanov [BI] and Dudley [D1,D2]. Gordon, Meyer, and Reisner [GMR1,GMR2] gave a constructive proof for the same inequality. Muller [Mü] showed that random approximation gives the same estimate. Gordon, Reisner, and Schutt [GRS] established the left hand inequality. Gruber [Gr2] obtained an asymptotic formula.

Our aim is to prove the following fact: PROPOSITION. Let K⊂ ℝn be a convex centrally symmetric body of volume 1, in isotropic position, i.e., Fix δ > 0 and choose m random points X l, … ,Xm ∈ K, where

Introduction. Let X be a random vector with log-concave distribution (for precise definitions see below). It is known that for any measurable seminorm and p, q > 0 the inequality holds with constants Cp,q depending only on p and q (see [4], Appendix III). In this paper we show that the above constants can be made independent of q, which is equivalent to the inequality

where IIXllo is the geometric mean of IIXII. In the particular case in which X is uniformly distributed on some convex compact set in Rn and the seminorm is given by some functional, inequality (1) was established by V. D. Milman and A. Pajor [3]. As a consequence of (1) we prove the result of Ullrich [6] concerning the equivalence of means for sums of independent Steinhaus random variables with vector coefficients, even though these random-variables are not log-concave (Corollary 2). To prove (1) we derive some estimates of log-concave measures of small balls (Corollary 1), which are of independent interest In the case of Gaussian random variables they were formulated and established in a weaker version in [5] and completelely proved in [2].

We say that a random vector X with values in E is log-concave if the distribution of X is log-concave. For a random vector X and a measurable seminorm ||.|| on E (i.e. Borel measurable, nonnegative, subadditive and positively homogeneous function on E) we define.

Let us begin with the following Lemma from [1].

The isotropic constant of a convex symmetric body K ⊂ ℝd is a highly important quantity. One of its equivalent definitions is where IKI stands for the volume of K and ||·||2 is the usual Euclidean norm. See [MP] for other formulations and a full discussion. Estimation of LK is one of the central problems on the border between local theory and convexity. Bourgain [Bo] has shown that LK ≲ d1/ 4 log d for any K ⊂ℝd, where by A ≲ B we mean that A ≲ cB for some universal constant c. He also raised the problem of universal boundedness of the isotropic constant independent of the dimension.

Till now there was no improvement of Bourgain's result in the general case, but boundedness of the isotropic constant was established for some families of bodies (see [MP,Ba,Jl,J2], for example), covering the unit balls of most of the classical spaces.

The case of the Schatten class spaces however, was left out. In [D] we showed that. Here we extend the result to all values of p using a simpler argument. First we shall recall the definiton of the Schatten class spaces.