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Every      m m      -dimensional subspace of      L p {L_p}      ,      1 &gt; p &gt; 2 1 &gt; p &gt; 2      ,      ( 1 + ε  ) (1 + \varepsilon )      -embeds into      l 1 n l_1^n       as long as      n ⩾  η  m 1 + ( 1 / p ) ( log ⁡  m ) −  1 n \geqslant \eta {m^{1 + (1/p)}}{(\log m)^{ - 1}}      , where      η  = η  ( p , ε  ) &gt; ∞  \eta = \eta (p,\varepsilon ) &gt; \infty      . For subspaces of      L 1 {L_1}       we get a somewhat weaker result.

Fine embeddings of finite-dimensional subspaces of 𝐿_{𝑝},1≤𝑝<2, into 𝑙^{𝑚}₁