Dvoretzky-type theorem for locally finite subsets of a Hilbert space

Catrina, Florin; Ostrovska, Sofiya; Ostrovskii, Mikhail  I.

doi:10.5802/aif.3672

Catrina, Florin ¹; Ostrovska, Sofiya ²; Ostrovskii, Mikhail I.¹

¹ Department of Mathematics and Computer Science, St. John’s University, 8000 Utopia Parkway, Queens, NY 11439 (USA)
² Department of Mathematics, Atilim University, 06830 Incek, Ankara (Turkey)

Annales de l'Institut Fourier, Online first, 43 p.

Abstract
Résumé

The main result of the paper: Given any $ε > 0$ , every locally finite subset of $ℓ_{2}$ admits a $(1 + ε)$ -bilipschitz embedding into an arbitrary infinite-dimensional Banach space. The result is based on two results which are of independent interest:

(1) A direct sum of two finite-dimensional Euclidean spaces contains a sub-sum of a controlled dimension which is $ε$ -close to a direct sum with respect to a $1$ -unconditional basis in a two-dimensional space.

(2) For any finite-dimensional Banach space $Y$ and its direct sum $X$ with itself with respect to a $1$ -unconditional basis in a two-dimensional space, there exists a $(1 + ε)$ -bilipschitz embedding of $Y$ into $X$ which on a small ball coincides with the identity map onto the first summand and on the complement of a large ball coincides with the identity map onto the second summand.

Le résultat principal de l’article : étant donné $ε > 0$ , chaque sous-ensemble localement fini de $ℓ_{2}$ admet un plongement $(1 + ε)$ -bilipschitz dans n’importe quel espace de Banach de dimension infinie. Le résultat est basé sur deux résultats qui présentent un intérêt indépendant :

(1) Une somme directe de deux espaces euclidiens de dimension finie contient une sous-somme de dimension contrôlée qui est $ε$ -proche d’une somme directe par rapport à une base $1$ -inconditionnelle dans un espace à deux dimensions.

(2) Pour tout espace de Banach de dimension finie $Y$ et sa somme directe $X$ avec lui-même par rapport à une base $1$ -inconditionnelle dans un espace à deux dimensions, il existe un plongement $(1 + ε)$ -bilipschitz de $Y$ dans $X$ qui coïncide, sur une petite boule, avec l’identité sur la première composante, et qui coïncide, sur le complément d’une grosse boule, avec l’identité sur la deuxième composante.

Received: 2022-03-07
Revised: 2023-01-12
Accepted: 2023-09-13
Online First: 2025-02-10

DOI: 10.5802/aif.3672

Classification: 46B85, 30L05, 46B07, 51F30
Keywords: Bilipschitz embedding, Dvoretzky Theorem, Finite-dimensional decomposition, Unconditional
Mots-clés : Plongement bilipschitz, Théorème de Dvoretzky, décomposition finie-dimensionnelle, base inconditionnelle

Author's affiliations:

Catrina, Florin ¹; Ostrovska, Sofiya ²; Ostrovskii, Mikhail I. ¹

¹ Department of Mathematics and Computer Science, St. John’s University, 8000 Utopia Parkway, Queens, NY 11439 (USA)
² Department of Mathematics, Atilim University, 06830 Incek, Ankara (Turkey)

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     author = {Catrina, Florin and Ostrovska, Sofiya and Ostrovskii, Mikhail  I.},
     title = {Dvoretzky-type theorem for locally finite subsets of a {Hilbert} space},
     journal = {Annales de l'Institut Fourier},
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     doi = {10.5802/aif.3672},
     language = {en},
     note = {Online first},
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Catrina, Florin; Ostrovska, Sofiya; Ostrovskii, Mikhail  I. Dvoretzky-type theorem for locally finite subsets of a Hilbert space. Annales de l'Institut Fourier, Online first, 43 p.

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