On L1-Embeddability of Unions of L1-Embeddable Metric Spaces and of Twisted Unions of Hypercubes

IF 0.9 3区数学 Q2 MATHEMATICS

Analysis and Geometry in Metric Spaces Pub Date : 2021-04-15 DOI:10.1515/agms-2022-0145

M. Ostrovskii, B. Randrianantoanina

引用次数: 3

Abstract

Abstract We study properties of twisted unions of metric spaces introduced in [Johnson, Lindenstrauss, and Schechtman 1986], and in [Naor and Rabani 2017]. In particular, we prove that under certain natural mild assumptions twisted unions of L1-embeddable metric spaces also embed in L1 with distortions bounded above by constants that do not depend on the metric spaces themselves, or on their size, but only on certain general parameters. This answers a question stated in [Naor 2015] and in [Naor and Rabani 2017]. In the second part of the paper we give new simple examples of metric spaces such that their every embedding into Lp, 1 ≤ p < ∞, has distortion at least 3, but which are a union of two subsets, each isometrically embeddable in Lp. This extends the result of [K. Makarychev and Y. Makarychev 2016] from Hilbert spaces to Lp-spaces, 1 ≤ p < ∞.

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关于L1可嵌入度量空间并集和超立方体扭曲并集的L1可嵌入性

摘要我们研究了[Johnson，Lindenstrauss，and Schechtman 1986]和[Naor and Rabani 2017]中引入的度量空间的扭并的性质。特别地，我们证明了在某些自然温和的假设下，L1可嵌入度量空间的扭曲并集也嵌入到L1中，其失真由不依赖于度量空间本身或其大小，而仅依赖于某些一般参数的常数所限定。这回答了[Naor 2015]和[Naor和Rabani 2017]中提出的问题。在本文的第二部分中，我们给出了度量空间的新的简单例子，使得它们在Lp，1≤p<∞中的每个嵌入都有至少3的失真，但它们是两个子集的并集，每个子集都可等距嵌入在Lp中。这将[K.Makarychev和Y.Makarychev2016]的结果从Hilbert空间扩展到Lp空间，1≤p<∞。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Analysis and Geometry in Metric Spaces Mathematics-Geometry and Topology

CiteScore

1.80

自引率

0.00%

发文量

审稿时长

16 weeks

期刊介绍： Analysis and Geometry in Metric Spaces is an open access electronic journal that publishes cutting-edge research on analytical and geometrical problems in metric spaces and applications. We strive to present a forum where all aspects of these problems can be discussed. AGMS is devoted to the publication of results on these and related topics: Geometric inequalities in metric spaces, Geometric measure theory and variational problems in metric spaces, Analytic and geometric problems in metric measure spaces, probability spaces, and manifolds with density, Analytic and geometric problems in sub-riemannian manifolds, Carnot groups, and pseudo-hermitian manifolds. Geometric control theory, Curvature in metric and length spaces, Geometric group theory, Harmonic Analysis. Potential theory, Mass transportation problems, Quasiconformal and quasiregular mappings. Quasiconformal geometry, PDEs associated to analytic and geometric problems in metric spaces.