Polylog dimensional subspaces of ℓ_∞^N

2018/12/10 by Gideon Schechtman, Schechtman, Gideon, Nicole Tomczak-Jaegermann +1
Mathematics · #46B07 #46B20 #FOS: Mathematics #Functional Analysis (math.FA) #Geometric and Algebraic Topology #Mathematical Dynamics and Fractals #Point processes and geometric inequalities

paper · pdf · doi:10.48550/arxiv.1812.03678

openalex publication_date 2018/12/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We show that a subspace of of ℓ_∞^N of dimension n>(log Nlog log N)² contains 2-isomorphic copies of ℓ_∞^k where k tends to infinity with n/(log Nlog log N)². More precisely, for every η>0, we show that any subspace of ℓ_∞^N of dimension n contains a subspace of dimension m=c(η)√(n)/(log Nlog log N) of distance at most 1+η from ℓ_∞^m.

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