Dvoretzky's Theorem Research Articles

On concentration, deviation and Dvoretzky's theorem for Besov, Lizorkin–Triebel and other spaces

We obtain explicit estimates of the constants related to the concentrations of measures and distance, deviation and Dvoretzky's theorem for the finite-dimensional subspaces of a wide class of function and other spaces including, in particular, various anisotropic spaces of Besov, Lizorkin–Triebel and Sobolev types endowed with geometrically friendly norms defined in terms of averaged differences, local polynomial approximations, functional calculus, wavelets and other means. New approaches are shown to be providing better estimates in the abstract setting as well.

On games with incomplete information and the Dvoretsky–Wald–Wolfowitz theorem with countable partitions

It has remained an open question as to whether the results of Milgrom–Weber [Milgrom, P.R., Weber, R.J., 1985. Distributional strategies for games with incomplete information. Mathematics of Operations Research 10, 619–632] are valid for action sets with a countably infinite number of elements without additional assumptions on the abstract measure space of information. In this paper, we give an affirmative answer to this question as a consequence of an extension of a theorem of Dvoretzky, Wald and Wolfowitz (henceforth DWW) due to Edwards [Edwards, D.A., 1987. On a theorem of Dvoretsky, Wald and Wolfowitz concerning Liapunov measures. Glasgow Mathematical Journal 29, 205–220]. We also present a direct elementary proof of the DWW theorem and its extension, one that may have an independent interest.

Purification and saturation

This paper illustrates the general technique established in 1984 by Hoover and Keisler for extending certain types of results from atomless Loeb measure spaces to measure spaces that we shall call countably generated. The Hoover-Keisler technique is applied here to further extend the authors' 2006 generalization of a theorem of Dvoretzky, Wald and Wolfowitz on the purification of measure-valued maps. The authors' 2006 result was first extended to these more general spaces by K. Podczeck in 2007; he used new results in functional analysis produced for that purpose. This paper demonstrates that, in general, such extensions follow from the Hoover-Keisler technique. Moreover, adaptations of counterexamples from earlier papers show that the extension obtained here holds only for nowhere countably generated spaces.

Konvexgeometrie

The meeting Konvexgeometrie organised by K. M. Ball, P. Goodey and P. M. Gruber, was held from December 17 to December 23, 2006. The meeting was attended by some 40 participants working in all areas of convex geometry. The program involved 10 plenary lectures of one hour's duration and about 15 shorter lectures. Some highlights of the program were as follows. Grigoris Paouris explained his proof that if K is an isotropic convex body of volume 1 in {\mathbf R}^n and the random variable X is uniformly distributed on K , then for some absolute constant C , \mathrm{P} (|X| > C \sigma t) \leq e^{-\sqrt{n} t} for all t > 1 , where \sigma^2 is the variance of X . The estimate is optimal apart from the value of C . Olivier Guédon then explained joint work with Fleury and Paouris, showing how the method of Paouris yields the central limit theorem for convex bodies, conjectured in 1996 by Ball and recently proved by Klartag. Assaf Naor described his new results with Manor Mendel which now give a complete picture of the non-linear Dvoretzky Theorem. 20 years ago, Bourgain, Figiel and Milman proved that any n -point metric space has a subset of size about \log n which can be embedded in Hilbert space with a constant distortion. In 2003, Bartal, Linial, Mendel and Naor discovered a remarkable threshold phenomenon: that if we allow distortion larger than a factor 2, there are subsets of size a power of n which are embeddable in Hilbert space. In the recent work the authors determine exactly the correct dependence of the power on the distortion: for each \epsilon >0 there are subsets of size n^{1-\epsilon} that are O(1/\epsilon) embeddable. Mark Rudelson described his recent estimates for the smallest singular values of almost square random (Gaussian) matrices. Considerably sharpening earlier work of Litvak, Pajor, himself, Tomczak and Vershynin and (in a slightly different direction) Candes and Tao, he established strong bounds for the probability that a random N \times n matrix maps a point of the unit sphere in {\mathbf R}^n to a point of small \ell_1^N norm. This is equivalent to understanding the maximum radius of an almost full-dimensional random section of the cross-polytope. The passage from such estimates to singular numbers uses standard techniques. Ralph Howard spoke about his recent work (joint with Paul Goodey) on bodies of constant brightness. This follows Howard's recent solution of the problem (dating back to 1926) of whether there exist bodies in 3-space which are of constant width and constant brightness but which are not Euclidean balls. Richard Gardner gave an account of several new algorithms for the reconstruction of convex bodies from their x-rays in a small number of directions. The new algorithms are robust in that they can accommodate noisy data and are sufficiently simple that convergence proofs are rendered quite straightforward. Matthias Reitzner gave a survey of the well-developed theory of random polytopes focussing on deviation estimates for the numbers of faces of given dimension. This covers work of himself, Bárány, Vu and others. Ryabogin and Zvavitch described their joint work with Nazarov solving a conjecture of Weil about the characterisation of zonoids. Semyon Alesker gave an impromptu evening presentation of the recent work of Greg Kuperberg who has given a very short and self-contained proof of the Bourgain-Milman theorem on the product of the volumes of a convex body and its polar. Mahler conjectured that this volume product is minimised by a simplex: the Bourgain–Milman theorem proves this up to a factor of (\textrm{constant})^n in n -dimensions which is what is needed for most applications.

Ramsey partitions and proximity data structures

This paper addresses two problems lying at the intersection of geometric analysis and theoretical computer science: The non-linear isomorphic Dvoretzky theorem and the design of good approximate distance oracles for large distortion. We introduce the notion of Ramsey partitions of a finite metric space, and show that the existence of good Ramsey partitions implies a solution to the metric Ramsey problem for large distortion (a.k.a. the non-linear version of the isomorphic Dvoretzky theorem, as introduced by Bourgain, Figiel, and Milman in [8]). We then proceed to construct optimal Ramsey partitions, and use them to show that for every \varepsilon \in (0,1) , every n -point metric space has a subset of size n^{1-\varepsilon} which embeds into Hilbert space with distortion O(1/\varepsilon) . This result is best possible and improves part of the metric Ramsey theorem of Bartal, Linial, Mendel and Naor [5], in addition to considerably simplifying its proof. We use our new Ramsey partitions to design approximate distance oracles with a universal constant query time, closing a gap left open by Thorup and Zwick in [32]. Namely, we show that for every n point metric space X , and k\geq 1 , there exists an O(k) -approximate distance oracle whose storage requirement is O\left( n^{1+1/k}\right) , and whose query time is a universal constant. We also discuss applications of Ramsey partitions to various other geometric data structure problems, such as the design of efficient data structures for approximate ranking.

Random ε-nets and embeddings in lN∞

We show that, given an n-dimensional normed space X, a sequence of N = (8/e) 2n independent random vectors (Xi) N=1, uniformly distributed in the unit ball of X � , with high probability forms an e-net for this unit ball. Thus the random linear map Γ : R n → R N defined by Γ x = (h x, Xii ) N=1 embeds X in l N with at most 1 + e norm distortion. In the case X = l n we obtain a random 1 + e-embedding into l N with asymptotically best possible relation between N, n, and e. 1. Introduction. Let X = (R n , kk ) be an arbitrary n-dimensional normed space with unit ball K. It is well known that, for any 0 < e < 1, X can be 1 + e-embedded into l N, for some N = N(e, n), depending on e and n, but independent of X. In this note we investigate 1+e-isomorphic embed- dings which are random with respect to some natural measure, depending on K. We first show that for N = (8/e) 2n , a sequence of N independent ran- dom vectors (Xi) N=1 , uniformly distributed in the unit ball K 0 of the dual space X ∗ , forms an e-net for K 0 with high probability. Thus, with high prob- ability, the random linear map Γ : R n → R N defined by Γ x = (h x, Xii ) N=1 embeds X in l N with at most 1 + e norm distortion. The important case is X = l n . In this case it is more natural to consider random vectors Xi uniformly distributed on the sphere S n−1 . Such vectors also form an e-net on the sphere, hence they determine a random 1 + e- embedding Γ of l n into l N. We also show that p n/N Γ is a 1+ e-isometry from l n into l N , with high probability. The case X = l n is connected with Dvoretzky's theorem ((D)). Milman found a new proof ((M)), using the Levy isoperimetric inequality on the

Zero sets of polynomials in several variables

Let $$k, n \in \mathbb{N}$$ , where n is odd. We show that there is an integer N = N(k, n) such that for every n-homogeneous polynomial $$P : \mathbb{R}^N \rightarrow \mathbb{R}$$ there exists a linear subspace $$X \hookrightarrow \mathbb{R}^N, \dim X = k$$ , such that P| X ≡ 0. This quantitative estimate improves on previous work of Birch et al., who studied this problem from an algebraic viewpoint. The topological method of proof presented here also allows us to obtain a partial solution to the Gromov-Milman problem (in dimension two) on an isometric version of a theorem of Dvoretzky.

On metric Ramsey-type phenomena

The main question studied in this article may be viewed as a nonlinear analogue of Dvoretzky's theorem in Banach space theory or as part of Ramsey theory in combinatorics. Given a finite metric space on n points, we seek its subspace of largest cardinality which can be embedded with a given distortion in Hilbert space. We provide nearly tight upper and lower bounds on the cardinality of this subspace in terms of n and the desired distortion. Our main theorem states that for any epsilon>0, every n point metric space contains a subset of size at least n^{1-\epsilon} which is embeddable in Hilbert space with O(\frac{\log(1/\epsilon)}{\epsilon}) distortion. The bound on the distortion is tight up to the log(1/\epsilon) factor. We further include a comprehensive study of various other aspects of this problem.

Two observations regarding embedding subsets of Euclidean spaces in normed spaces

This paper contains two results concerning linear embeddings of subsets of Euclidean space in low-dimensional normed spaces. The first is an improvement of the known dependence on ɛ in Dvoretzky's theorem from order of ɛ 2 to order of ɛ (except for log factors). The second is a joint generalization of (Milman's version of) Dvoretzky's theorem and (a recent generalization by Klartag and Mendelson of) the Johnson–Lindenstrauss Lemma.

Planar Sections of Convex Bodies and Universal Fibrations

A conjecture on tautological vector bundles over Grassmannians, which generalizes the well-known Dvoretzky theorem, is stated, discussed, and also proved in one nontrivial case: for the Grassmannian of 2-planes. It is also proved that each three-dimensional real normed space contains a two-dimensional subspace with Banach―Mazur distance from the Euclidean plane at most $$\frac{1}{2}\log (4/3)$$ and this estimate is sharp. Bibliography: 12 titles.

Randomized isomorphic Dvoretzky theorem

Let K be a symmetric convex body in R N for which B 2 N is the ellipsoid of minimal volume. We provide estimates for the geometric distance of a ‘typical’ rank n projection of K to B 2 n , for 1⩽ n< N. Known examples show that the resulting estimates are optimal (up to numerical constants) even for the Banach–Mazur distance. To cite this article: A. Litvak et al., C. R. Acad. Sci. Paris, Ser. I 335 (2002) 345–350.

An isomorphic Dvoretzky's theorem for convex bodies

We prove that there exist constants C>0 and 0 < λ < 1 so that for all convex bodies K in $ℝ^n$ with non-empty interior and all integers k so that 1 ≤ k ≤ λn/ln(n+1), there exists a k-dimensional affine subspace Y of $ℝ^n$ satisfying $d(Y ∩ K, B_2^k) ≤ C(1

A Dvoretzky Theorem for Polynomials

A Dvoretzky Theorem for Polynomials

Dvoretzky's theorem — Thirty years later

Dvoretzky's theorem — Thirty years later

Dvoretzky's theorem for quasi-normed spaces

Dvoretzky's theorem for quasi-normed spaces

The dimension of Euclidean subspaces of quasi-normed spaces

The purpose of this article is to extend certain results which are known to hold for convex bodies to a class of non-convex bodies occurring in the theory of topological vector spaces. In the first section after this introduction an analogue of F. John's Theorem on the distance of a finite dimensional space from Euclidean space is obtained, and the result is shown to be best possible. The Dvoretzky-Rogers Lemma on the points of contact of a symmetric convex body with the ellipsoid of maximum volume contained within it is discussed for certain non-convex bodies. In the next part the Dvoretzky Theorem on the existence of ellipsoidal sections is shown to hold with the best possible estimate for the dimension of the sections. It follows from estimates involving cotype constants that the finite dimensional subspaces of Lp (0 &lt; p &lt; 1) possess large almost Hilbertian subspaces. The final section extends the theorem of S. Szarek relating the volume of a body to the existence of ellipsoidal sections.

Absolutely divergent series and classes of Banach spaces

A normed space E is said to be series immersed in a Banach space X if for every absolutely divergent series nxn in E there is a continuous linear mapping T from E into X such that nTxn diverges absolutely. The theorem of Dvoretzky and Rogers(1) implies that a normed space E is series immersed in a finite dimensional space if and only if E itself is finite dimensional. In (4) and (9) it was shown that E is series immersed in lp 1 p &lt; if and only if E is isomorphic to a subspace of Lp() for some measure . In particular, E is isomorphic to an inner product space if and only if it is series immersed in a Hilbert space. The property of series immersion was further studied in the papers (7) and (8). The main results in these two papers are conditions on X under which E series immersed in X would imply E locally immersed in X, a condition slightly stronger (formally) than E finitely representable in X.

Holomorphic Semi-Groups and the Geometry of Banach Spaces

According to a fundamental theorem of Dvoretzky (cf. [24], [25]), any infinite dimensional Banach space X contains for each integer n and each E > 0 a subspace Xn which is (1 + e)-isomorphic to 12n) In the classical Banach spaces Lp, and 1p, with 1 < p < x, it is well known that this result can be improved: These spaces contain complemented ln)7's (this means that we can find subspaces Xn as above with the additional property that there exist projections P.: X -* Xn such that supn 11 Pn 11 < cx). Moreover, it is also well known that this property no longer holds for 11 or lo. This motivated the question (cf. [13]) to determine which infinite dimensional Banach spaces contain complemented 1(n)'s. We will prove below that all uniformly convex spaces have this property. Moreover, we will show that a Banach space X verifies this property locally (see definition 2.10 for more precision) if and only if X does not contain l(n)'s uniformly. Progressively, the work of previous authors (cf. [14], [12] and particularly [5]) has reduced such so-called geometric questions to a purely analytic problem consisting of verifying in X a certain inequality referred to as K-convexity. In this paper, we connect this inequality with the holomorphy of a certain semi-group of operators on Lp(X) and we show how this leads to the results already mentioned.

Learning extremal regulator implementation by a stochastic automaton and stochastic approximation theory

There exist many different approaches to the investigation of the characteristics of learning system. These approaches use different branches of mathematics and, thus, obtain different results, some of them are too complicated and others do not match the results of practical experiments. This paper presents the modelling of learning systems by means of stochastic automate, mainly one particular model of a learning extremal regulator. The proof of convergence is based on Dvoretzky's Theorem on stochastic approximations. Experiments have proved the theory of stochastic automata and stochastic approximations to be quite suitable means for studying the learning systems.

Explicit bounds for the departure from normality of sums of dependent random variables

In this paper, the substitution methods of DVORETZKY [4] are used to bound the departure from normality of a large class of processes which are martingales, or at least close to being martingales. Two main results are given. Theorem 2.1 generalizes Theorem 1 of BASU [1] in the following ways: the function f is not assumed to have a third derivative, the conditional expectations are not assumed to vanish, and the sum of the conditional variances is not assumed to be almost surely unity. As a corollary we are able to give a uniform bound on A (Sn, N) which goes to zero for any array satisfying the conditions of Dvoretzky's main theorem ([4], Theorem 2.2). However, our result cannot give a rate of convergence for all systems satisfying the conditions of Corollary (2.7) of McLEISH [7]. On the other hand McLeish's result does not contain that of Dvoretzky as he claims. For if we take X,i, i = l , . . . , n independent with