Almost-isometric Gaussian embedding from l2 into l1 (${\ell }_2^n\hookrightarrow {\ell }_1^k$)

The random matrixsatisfiesA net argument shows that $k\ge C_{\epsilon }n$ permits a linear embedding of distortion below $1+\epsilon$.

Every $n$-point metric space embeds into ${\ell }_1^{O(\log n)}$ with distortion $O(\log n)$. Apply the Bourgain embedding theorem, reduce its Euclidean dimension to $O(\log n)$ with the Johnson–Lindenstrauss lemma, and use an Almost-isometric Gaussian embedding from l2 into l1.