W. Kandinsky: Accent on Rose, 1926

Embedding \(\ell_1^d\) into \(\ell_2^d\)

Part 1. Prove that there exists an embedding of \(\ell_1^d\) into \(\ell_2^d\) with distortion \(\sqrt{d}\).

Part 2\(^*\). Prove that every map \(f\) from \(\ell_1^d\) into \(\ell_2^d\) has ditortion at least \(\sqrt{d}\). To this end, show the following.
  1. Assume first that \(f\) is a linear map. Consider the standard basis \(e_1,\dots, e_d\) of \(\ell_1^d\). Let \(r_1, \dots, r_d\in\{\pm 1\}\) be independent unbiased Bernoulli random variables. Choose an index \(j\in\{1,\dots, d\}\) uniformly at random. Define \(r'_i\) by \(r'_i = r_i\) if \(i\neq j\), and \(r'_i = -r_i\) if \(i= j\). (That is, \(r'_i = r_i\) for all but one index \(i\).) Let r.v. \(u = \sum_{i=1}^d r_i e_i\) and \(u' = \sum_{i=1}^d r_i' e_i\). Compute the values of $$\frac{{\mathbb E}{\|u - u'\|_1^2}}{{\mathbb E}{\|u - (-u)\|_1^2}} \quad\text{and}\quad \frac{{\mathbb E}{\|f(u) - f(u')\|_2^2}}{{\mathbb E}{\|f(u) - f(-u)\|_2^2}}.$$ Conclude that \(f\) has ditortion at least \(\sqrt{d}\).
  2. By Rademacher's theorem, every Lipschitz map from \({\mathbb R}^n\) to \({\mathbb R}^m\) is differentiable almost everywhere. Let \(f\) be a Lipschitz map from \(\ell_1^d\) to \(\ell_2^d\). Apply Rademacher's theorem to \(f\) and consider its differential \(df\) at some point. Using that \(df\) is a linear map, bound its distortion from below. Prove that \(f\) has distortion at least \(\sqrt{d}\).