 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 $$\sqrt{d}$$.