Uniform embeddings of hyperbolic groups in hilbert spaces

We construct uniform embeddings of the Cayley graphs of hyperbolic groups and cyclic extensions of torsion-free small cancellation groups in Hilbert spaces. Supported by a Rothschild postdoctoral followship and the IHES.     

Finite metric spaces needing high dimension for lipschitz embeddings in banach spaces

We construct a sequence of metric spaces (M _n) with cardM _n=3n satisfying that for everyc<2, there exists a real numbera(c)>0 such that, if the Lipschitz distance fromM _n to a subset of a Banach spaceE is less thanc, then dim(E) ≥a(c)n. We also prove several results about embeddings of metric spaces whose non-zero distance values are in the interval [1,2].     

Coarse embeddings of metric spaces into Banach spaces

There are several characterizations of coarse embeddability of locally finite metric spaces into a Hilbert space. In this note we give such characterizations for general metric spaces. By applying these results to the spaces , we get their coarse embeddability into a Hilbert space for . This together with a theorem by Banach and Mazur yields that coarse embeddability into and into are equivalent when . A theorem by G.Yu and the above allow us to extend to , , the range of spaces, coarse embeddings into which is guaranteed for a finitely generated group to satisfy the Novikov Conjecture.

     

Remarks on coarse embeddings of metric spaces into uniformly convex Banach spaces

We study the support and convergence conditions for a metric space to be coarsely embeddable into a uniformly convex Banach space. By using ultraproducts we also show that the coarse embeddability of a metric space into a uniformly convex Banach space is determined by its finite subspaces.     

Uniform embeddings of Banach spaces

It is proved that for 1<-p≤2,L _p(0,1) andl _p are uniformly equivalent to bounded subsets of themselves. It is also shown that for 1<=p<=2, 1≦q<∞,L _p is uniformly equivalent to a subset ofl _q. This is a part of the author’s Ph. D. thesis prepared at the Hebrew University of Jerusalem under the supervision of Professor J. Lindenstrauss. The author wishes to thank Professor Lindenstrauss for his guidance.     

Low distortion embeddings of some metric graphs into Banach spaces

We give a simple example of a countable metric graph M such that M Lipschitz embeds with distortion strictly less than 2 into a Banach space X only if X contains an isomorphic copy of l ₁. Further we show that, for each ordinal α < ω ₁, the space C([0, ω ^α ]) does not Lipschitz embed into C(K) with distortion strictly less than 2 unless K ^(α) ≠ 0. Also \(C\left( {\left[ {0,{\omega ^{{\omega ^\alpha }}}} \right]} \right)\) does not Lipschitz embed into a Banach space X with distortion strictly less than 2 unless Sz(X) ≥ ω ^α+1.     

Isometric embeddings of finite metric spaces

It is proved that there exists a metric on a Cantor set such that any finite metric space whose diameter does not exceed 1 and the number of points does not exceed n can be isometrically embedded into it. It is also proved that for any m, n ∈ N there exists a Cantor set in R^m that isometrically contains all finite metric spaces which can be embedded into R^m, contain at most n points, and have the diameter at most 1. The latter result is proved for a wide class of metrics on R^m and, in particular, for the Euclidean metric.     

Euclidean embeddings of finite metric spaces

This is a brief survey on Euclidean embeddings of finite metric spaces, focusing on the power transform metric with many examples. Some old results are presented in slightly improved forms, and the last section contains a few new results. Proofs are given if they are elementary and not too long. Several problems and conjectures are also given.     

Uniform embeddings of bounded geometry spaces into reflexive Banach space

We show that every metric space with bounded geometry uniformly embeds into a direct sum of spaces ('s going off to infinity). In particular, every sequence of expanding graphs uniformly embeds into such a reflexive Banach space even though no such sequence uniformly embeds into a fixed space. In the case of discrete groups we prove the analogue of a--menability - the existence of a metrically proper affine isometric action on a direct sum of spaces.

     

Lipschitz and path isometric embeddings of metric spaces

We prove that each sub-Riemannian manifold can be embedded in some Euclidean space preserving the length of all the curves in the manifold. The result is an extension of Nash C ¹ Embedding Theorem. For more general metric spaces the same result is false, e.g., for Finsler non-Riemannian manifolds. However, we also show that any metric space of finite Hausdorff dimension can be embedded in some Euclidean space via a Lipschitz map.     

Embedding unconditional stable banach spaces into symmetric stable banach spaces

In this paper we prove the following result which solves a question raised by A. Pelczynski: “Every stable Banach space with an unconditional basis is isomorphic to a complemented subspace of some stable Banach space with a symmetric basis.” Moreover, we show that all the interpolation spacesl ^p ,l ^q _θ,X,1 1≦p, q<∞ andX stable, are stable.     

A general metric regularity in asplund banach spaces

This paper establishes a simple and easily-applied criterion for determining whether a multivalued mapping is metrically regular relatively to a subset in the range space.     

A continuity criterion for metric projections in banach spaces

The paper provides the complete characteristics of the class of reflexive strictly convex spaces in which the metric projections on each convex closed set are continuous.Translated from Matematicheskie Zametki, Vol. 10, No. 4, pp. 459–468, October, 1971.The author wishes to thank S. B. Stechkin for useful comments.     

Dense embeddings of nowhere locally compact separable metric spaces

Nowhere locally compact separable metric spaces are characterized (up to homeomorphism) as precisely the dense subsets of the separable Hilbert space and those of dimension at most n are characterized (up to homeomorphism) as precisely the dense subsets of the n-dimensional Menger-Nöbeling space.     

Sobolev embeddings in metric measure spaces with variable dimension

In this article we study metric measure spaces with variable dimension. We consider Lebesgue spaces on these sets, and embeddings of the Riesz potential in these spaces. We also investigate Hajłasz-type Sobolev spaces, and prove Sobolev and Trudinger inequalities with optimal exponents. All of these questions lead naturally to function spaces with variable exponents. Supported the Research Council of Norway, Project 160192/V30.     

Dimension estimate preserving embeddings for compacta in metric spaces

We generalize an embedding result for compacta with finite fractal dimension, which is due to B. R. Hunt and V. Yu. Kaloshin [3] for Banach spaces, to metric spaces and derive estimates for the fractal dimension of the embedded compact set.