In this paper, we study the coarse embedding into Banach space. We proved that under certain conditions, the property of embedding into Banach space can be preserved under taking the union the metric spaces. For a group $G$ strongly relative hyperbolic to a subgroup $H$, we proved that if $H$ admits a coarse embedding into a uniformly convex Banach space, so is $B(n)=\{g\in G|\abs{g}_{S\cup\mathscr{H}}\leq n\}$.

2011-05-17math.FAmath.MG

Embeddings of proper metric spaces into Banach spaces

We show that there exists a strong uniform embedding from any proper metric space into any Banach space without cotype. Then we prove a result concerning the Lipschitz embedding of locally finite subsets of $\mathcal{L}_{p}$-spaces. We use this locally finite result to construct a coarse bi-Lipschitz embedding for proper subsets of any $\mathcal{L}_p$-space into any Banach space $X$ containing the $\ell_p^n$'s. Finally using an argument of G. Schechtman we prove that for general proper metric spaces and for Banach spaces without cotype a converse statement holds.

2009-06-19math.FAmath.MG

Isometric embeddings of a class of separable metric spaces into Banach spaces

Let $(M,d)$ be a bounded countable metric space and $c>0$ a constant, such that $d(x,y)+d(y,z)-d(x,z) \ge c$, for any pairwise distinct points $x,y,z$ of $M$. For such metric spaces we prove that they can be isometrically embedded into any Banach space containing an isomorphic copy of $\ell_\infty$.

2018-02-27math.FAmath.MG

Metric spaces admitting low-distortion embeddings into all $n$-dimensional Banach spaces

For a fixed $K\gg 1$ and $n\in\mathbb{N}$, $n\gg 1$, we study metric spaces which admit embeddings with distortion $\le K$ into each $n$-dimensional Banach space. Classical examples include spaces embeddable into $\log n$-dimensional Euclidean spaces, and equilateral spaces. We prove that good embeddability properties are preserved under the operation of metric composition of metric spaces. In particular, we prove that any $n$-point ultrametric can be embedded with uniformly bounded distortion into any Banach space of dimension $\log n$. The main result of the paper is a new example of a famil…

2015-08-18math.FAmath.MG

Coarse embeddings of metric spaces into Banach spaces

There are several characterizations of coarse embeddability of a discrete metric space into a Hilbert space. In this note we give such characterizations for general metric spaces. By applying these results to the spaces $L_p(μ)$, we get their coarse embeddability into a Hilbert space for $0<p<2$. This together with a theorem by Banach and Mazur yields that coarse embeddability into $\ell_2$ and into $L_p(0,1)$ are equivalent when $1 \le p<2$. A theorem by G.Yu and the above allow to extend to $L_p(μ)$, $0<p<2$, the range of spaces, coarse embedding into which guarantees for a finitely generate…

2004-04-22math.FAmath.MG

Distortion in the finite determination result for embeddings of locally finite metric spaces into Banach spaces

Given a Banach space $X$ and a real number $α\ge 1$, we write: (1) $D(X)\leα$ if, for any locally finite metric space $A$, all finite subsets of which admit bilipschitz embeddings into $X$ with distortions $\le C$, the space $A$ itself admits a bilipschitz embedding into $X$ with distortion $\le α\cdot C$; (2) $D(X)=α^+$ if, for every $\varepsilon>0$, the condition $D(X)\leα+\varepsilon$ holds, while $D(X)\leα$ does not; (3) $D(X)\le α^+$ if $D(X)=α^+$ or $D(X)\le α$. It is known that $D(X)$ is bounded by a universal constant, but the available estimates for this constant are rather large. The…

2017-12-20math.FAmath.MG

更多相关论文收起更多论文

A coding of bundle graphs and their embeddings into Banach spaces

The purpose of this article is to generalize some known characterizations of Banach space properties in terms of graph preclusion. In particular, it is shown that superreflexivity can be characterized by the non-equi-bi-Lipschitz embeddability of any family of bundle graphs generated by a nontrivial finitely-branching bundle graph. It is likewise shown that asymptotic uniform convexifiability can be characterized within the class of reflexive Banach spaces with an unconditional asymptotic structure by the non-equi-bi-Lipschitz embeddability of any family of bundle graphs generated by a nontriv…

2017-10-02math.FAmath.MG

On metric characterizations of some classes of Banach spaces

The paper contains the following results and observations: (1) There exists a sequence of unweighted graphs $\{G_n\}_n$ with maximum degree 3 such that a Banach space $X$ has no nontrivial cotype iff $\{G_n\}_n$ admit uniformly bilipschitz embeddings into $X$; (2) The same for Banach spaces with no nontrivial type; (3) A sequence $\{G_n\}$ characterizing Banach spaces with no nontrivial cotype in the sense described above can be chosen to be a sequence of bounded degree expanders; (4) The infinite diamond does not admit a bilipschitz embedding into Banach spaces with the Radon-Nikodým property…

2011-02-24math.FAmath.MG

A new approach to low-distortion embeddings of finite metric spaces into non-superreflexive Banach spaces

The main goal of this paper is to develop a new embedding method which we use to show that some finite metric spaces admit low-distortion embeddings into all non-superreflexive spaces. This method is based on the theory of equal-signs-additive sequences developed by Brunel and Sucheston (1975-1976). We also show that some of the low-distortion embeddability results obtained using this method cannot be obtained using the method based on the factorization between the summing basis and the unit vector basis of $\ell_1$, which was used by Bourgain (1986) and Johnson and Schechtman (2009).

2017-02-23math.FAmath.MG

On metric characterizations of the Radon-Nikodým and related properties of Banach spaces

We find a class of metric structures which do not admit bilipschitz embeddings into Banach spaces with the Radon-Nikodým property. Our proof relies on Chatterji's (1968) martingale characterization of the RNP and does not use the Cheeger's (1999) metric differentiation theory. The class includes the infinite diamond and both Laakso (2000) spaces. We also show that for each of these structures there is a non-RNP Banach space which does not admit its bilipschitz embedding. We prove that a dual Banach space does not have the RNP if and only if it admits a bilipschitz embedding of the infinite dia…

2013-02-24math.FAmath.MG

Metrical characterization of super-reflexivity and linear type of Banach spaces

We prove that a Banach space X is not super-reflexive if and only if the hyperbolic infinite tree embeds metrically into X. We improve one implication of J.Bourgain's result who gave a metrical characterization of super-reflexivity in Banach spaces in terms of uniforms embeddings of the finite trees. A characterization of the linear type for Banach spaces is given using the embedding of the infinite tree equipped with a suitable metric.

2007-04-16math.FAmath.MG

Remarks on quasi-isometric non-embeddability into uniformly convex Banach spaces

We construct a locally finite graph and a bounded geometry metric space which do not admit a quasi-isometric embedding into any uniformly convex Banach space. Connections with the geometry of $c_0$ and superreflexivity are discussed.

2015-01-28math.FAmath.MG

嵌入 Banach 空间的有限确定性：新证明与低失真

Finite determination for embeddings into Banach spaces: new proof, low distortion

本文改进了 Ostrovskii 关于局部有限度量空间双 Lipschitz 与粗嵌入 Banach 空间的有限确定性结果，通过引入对数螺线粘合大幅降低失真至 3+ε，并利用 Brunel-Sucheston 扩展模型存在性结果简化了证明。

2023-07-27math.FAmath.MG

The geometry of Hamming-type metrics and their embeddings into Banach spaces

Within the class of reflexive Banach spaces, we prove a metric characterization of the class of asymptotic-$c_0$ spaces in terms of a bi-Lipschitz invariant which involves metrics that generalize the Hamming metric on $k$-subsets of $\mathbb{N}$. We apply this characterization to show that the class of separable, reflexive, and asymptotic-$c_0$ Banach spaces is non-Borel co-analytic. Finally, we introduce a relaxation of the asymptotic-$c_0$ property, called the asymptotic-subsequential-$c_0$ property, which is a partial obstruction to the equi-coarse embeddability of the sequence of Hamming g…

2020-04-09math.FAmath.MG

将Wasserstein空间的大子集嵌入Banach空间

Embedding large subsets of Wasserstein spaces into Banach spaces

Université Paris-Saclay（巴黎萨克雷大学）；CNRS（法国国家科学研究中心）；Inria（法国研究与数字联盟）；Laboratoire de mathématiques d’Orsay（奥赛数学实验室）；École normale supérieure（巴黎高等师范学院）；Université PSL（巴黎文理研究大学）

本研究证明三种经典嵌入在带矩界的Wasserstein空间子集上为双Hölder映射，其结果与完整Wasserstein空间嵌入非平凡型Banach空间的已知阻碍形成对比，还明确了结论成立所需矩的尖锐性。

2026-10-09math.MG

Characterization of quasi-Banach spaces which coarsely embed into a Hilbert space

A map f between two metric spaces (X,d_1) and (Y,d_2) is called a coarse embedding of X into Y if there exist two nondecreasing functions phi_1, phi_2:[0,\infty) --> [0,\infty) such that: phi_1(d_1(x,y)) \leq d_2(f(x),f(y)) \leq phi_2(d_1(x,y)) for all x, y in X, and phi_1(t) tends to \infty as t tends to \infty. We characterize those quasi-Banach spaces that have a coarse embedding into a Hilbert space.

2004-11-11math.FAmath.MG

On Kalton's interlaced graphs and nonlinear embeddings into dual Banach spaces

We study the nonlinear embeddability of Banach spaces and the equi-embeddability of the family of Kalton's interlaced graphs $([\mathbb N]^k,d_{\mathbb K})_k$ into dual spaces. Notably, we define and study a modification of Kalton's property $\mathcal Q$ that we call property $\mathcal{Q}_p$ (with $p \in (1,+\infty]$). We show that if $([\mathbb N]^k,d_{\mathbb K})_k$ equi-coarse Lipschitzly embeds into $X^*$, then the Szlenk index of $X$ is greater than $ω$, and that this is optimal, i.e., there exists a separable dual space $Y^*$ that contains $([\mathbb N]^k,d_{\mathbb K})_k$ equi-Lipschitz…

2021-03-01math.FAmath.MG

On embeddings of finite subsets of $\ell_2$

We study finite subsets of $\ell_2$, and more generally any metric space, and consider whether these isometrically embed into a Banach space. Our results partially answer a question of Ostrovskii, on whether every infinite-dimensional Banach space contains every finite subset of $\ell_2$ isometrically. The updated version contains acknowledgement that Theorem 3.1 has been proven previously in a paper of Shkarin.

2016-09-29math.FAmath.MG

Different forms of metric characterizations of classes of Banach spaces

For each sequence X of finite-dimensional Banach spaces there exists a sequence H of finite connected nweighted graphs with maximum degree 3 such that the following conditions on a Banach space Y are equivalent: (1) Y admits uniformly isomorphic embeddings of elements of the sequence X. (2) Y admits uniformly bilipschitz embeddings of elements of the sequence H.

2011-12-04math.COmath.FAmath.MG

Isometric embeddings of snowflakes into finite-dimensional Banach spaces

We consider a general notion of snowflake of a metric space by composing the distance by a nontrivial concave function. We prove that a snowflake of a metric space $X$ isometrically embeds into some finite-dimensional normed space if and only if $X$ is finite. In the case of power functions we give a uniform bound on the cardinality of $X$ depending only on the power exponent and the dimension of the vector space.

2016-09-12math.MG

Fibred cofinitely-coarse embeddability of box families and proper isometric affine actions on uniformly convex Banach spaces

In this paper we show that a countable, residually amenable group admits a proper isometric affine action on some uniformly convex Banach space if and only if one (or equivalently, all) of its box families admits a fibred cofinitely-coarse embedding into some uniformly convex Banach space.

2017-10-23math.FAmath.MG

关于可数分支图的类 Matoušek 嵌入障碍

On Matoušek-like Embedding Obstructions of Countably Branching Graphs

本文利用受 Matoušek 启发的纯度量方法，给出了可数分支树与菱形不可嵌入特定 Banach 空间的新证明，将其推广至满足类曲率不等式的度量空间，并推广 Tessera 的嵌入结果给出了 Lipschitz 嵌入的压缩下界。

2023-10-08math.FAmath.MG

On the finite dimensional approximation of the Kuratowski-embedding for compact manifolds

In the proof of his systolic inequality, Gromov uses an isometric embedding of a Riemannian manifold M into the Banach space of bounded functions on M, the so-called Kuratowski-embedding. Subsequently, it was shown by different authors that the Kuratowski embedding can be approximated by bi-Lipschitz embeddings into finite-dimensional Banach spaces. We give a detailed proof for the existence of such finite-dimensional approximations along the lines suggested by Larry Guth and go on to discuss quantitative aspects of the problem, establishing for the dimension of the Banach space a bound which …

2013-07-03math.MG

Test-space characterizations of some classes of Banach spaces

Let $\mathcal{P}$ be a class of Banach spaces and let $T=\{T_α\}_{α\in A}$ be a set of metric spaces. We say that $T$ is a set of {\it test-spaces} for $\mathcal{P}$ if the following two conditions are equivalent: (1) $X\notin\mathcal{P}$; (2) The spaces $\{T_α\}_{α\in A}$ admit uniformly bilipschitz embeddings into $X$. The first part of the paper is devoted to a simplification of the proof of the following test-space characterization obtained in M.I. Ostrovskii [Different forms of metric characterizations of classes of Banach spaces, Houston J. Math., to appear]: For each sequence $\{X_m\}_{…

2011-12-14math.COmath.FAmath.MG

Embedding fractals in Banach, Hilbert or Euclidean spaces

By a metric fractal we understand a compact metric space $K$ endowed with a finite family $\mathcal F$ of contracting self-maps of $K$ such that $K=\bigcup_{f\in\mathcal F}f(K)$. If $K$ is a subset of a metric space $X$ and each $f\in\mathcal F$ extends to a contracting self-map of $X$, then we say that $(K,\mathcal F)$ is a fractal in $X$. We prove that each metric fractal $(K,\mathcal F)$ is $\bullet$ isometrically equivalent to a fractal in the Banach spaces $C[0,1]$ and $\ell_\infty$; $\bullet$ bi-Lipschitz equivalent to a fractal in the Banach space $c_0$; $\bullet$ isometrically equivale…

2018-12-14math.GNmath.MG