The main result of the paper: Given any $\varepsilon >0$, every locally finite subset of $\ell _2$ admits a $(1+\varepsilon )$-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 $\varepsilon $-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 $(1+\varepsilon )$-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 $\varepsilon >0$, chaque sous-ensemble localement fini de $\ell _2$ admet un plongement $(1+\varepsilon )$-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 $\varepsilon $-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 $(1+\varepsilon )$-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.
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Keywords: Bilipschitz embedding, Dvoretzky Theorem, Finite-dimensional decomposition, Unconditional
Mots-clés : Plongement bilipschitz, Théorème de Dvoretzky, décomposition finie-dimensionnelle, base inconditionnelle
Catrina, Florin 1; Ostrovska, Sofiya 2; Ostrovskii, Mikhail I. 1
CC-BY-ND 4.0
@article{AIF_2025__75_6_2565_0,
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},
pages = {2565--2607},
year = {2025},
publisher = {Association des Annales de l{\textquoteright}institut Fourier},
volume = {75},
number = {6},
doi = {10.5802/aif.3672},
language = {en},
url = {https://aif.centre-mersenne.org/articles/10.5802/aif.3672/}
}
TY - JOUR AU - Catrina, Florin AU - Ostrovska, Sofiya AU - Ostrovskii, Mikhail I. TI - Dvoretzky-type theorem for locally finite subsets of a Hilbert space JO - Annales de l'Institut Fourier PY - 2025 SP - 2565 EP - 2607 VL - 75 IS - 6 PB - Association des Annales de l’institut Fourier UR - https://aif.centre-mersenne.org/articles/10.5802/aif.3672/ DO - 10.5802/aif.3672 LA - en ID - AIF_2025__75_6_2565_0 ER -
%0 Journal Article %A Catrina, Florin %A Ostrovska, Sofiya %A Ostrovskii, Mikhail I. %T Dvoretzky-type theorem for locally finite subsets of a Hilbert space %J Annales de l'Institut Fourier %D 2025 %P 2565-2607 %V 75 %N 6 %I Association des Annales de l’institut Fourier %U https://aif.centre-mersenne.org/articles/10.5802/aif.3672/ %R 10.5802/aif.3672 %G en %F AIF_2025__75_6_2565_0
Catrina, Florin; Ostrovska, Sofiya; Ostrovskii, Mikhail I. Dvoretzky-type theorem for locally finite subsets of a Hilbert space. Annales de l'Institut Fourier, Volume 75 (2025) no. 6, pp. 2565-2607. doi: 10.5802/aif.3672
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