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

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

doi:10.5802/aif.3672

Dvoretzky-type theorem for locally finite subsets of a Hilbert space
[Un théorème à la Dvoretzky pour les sous-ensembles localement finis d’un espace de Hilbert]

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.

Résumé
Abstract

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.

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.

Reçu le : 2022-03-07
Révisé le : 2023-01-12
Accepté le : 2023-09-13
Première publication : 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

Affiliations des auteurs :

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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     title = {Dvoretzky-type theorem for locally finite subsets of a {Hilbert} space},
     journal = {Annales de l'Institut Fourier},
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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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