Embedding subspaces of 𝐿₁ into 𝑙^{𝑁}₁

Talagrand, Michel

doi:10.1090/S0002-9939-1990-0994792-4

Embedding subspaces of $L_ 1$ into $l^ N_ 1$
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by Michel Talagrand

Proc. Amer. Math. Soc. 108 (1990), 363-369

DOI: https://doi.org/10.1090/S0002-9939-1990-0994792-4

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Abstract:

We simplify techniques of Schechtman, Bourgain, Lindenstrauss, Milman, to prove the following. If $X$ is an $n$-dimensional subspace of ${L_1}$, there exists a subspace $Y$ of $l_1^N$ such that $d\left ( {X,Y} \right ) \leq 1 + \varepsilon$ whenever $N \geq CK{\left ( X \right )^2}{\varepsilon ^{ - 2}}n$, where $K\left ( X \right )$ is the $K$-convexity constant of $X$, and where $C$ is a universal constant.

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