Very tight embeddings of subspaces of L _p , 1 ≤ p < 2, into l ⁿ _p

Abstract

We prove that for 1 ≤ p < r < 2, every n-dimensional subspace E of L _r, in particular l ⁿ _r , well-embeds into l ^m _p for some m ≤ (1 + $$\epsilon$$)n, where “well” depends on p, r, and the arbitrary $$\epsilon$$ > 0, but not on n.