Lipschitz functions that saturate the Lipschitz inequality on the average (part 1)

Question

Consider a 1-Lipschitz function $f: \mathbb R^n \to \mathbb R$ satisfying the inequality \begin{align*} |f(x) - f(y)| \le \|x-y\|_2, \;\forall x,y \in \mathbb R^n. \end{align*} For $n \ge 2$, can we find a 1-Lipschitz function that saturates the above inequality on the average?

To make the notion of "on the average" precise, let $x$ and $y$ be independent standard Gaussian vectors, i.e., $x,y \sim N(0,I_n)$. One can show that $$\mathbb E \big|\|x\|_2 - \|y\|_2\big| \asymp 1.$$ while $$\mathbb E\|x-y\|_2 \asymp \sqrt{n}.$$ Is there a 1-Lipschitz function $f : \mathbb R^n \to \mathbb R$ such that $$\mathbb E|f(x) - f(y)| \asymp \sqrt{n}?$$ Here $\asymp$ means inequalities go in both directions up to constants.

A related question is determining the order of $$ \sup_{f \in \text{Lip}(1)}\mathbb E|f(x) - f(y)| $$ where $\text{Lip}(1)$ is the set of $1$-Lipschitz functions from $\mathbb R^n$ to $\mathbb R$.

Answer 1

There is no $1$-Lipschitz function $f\colon \mathbb R^n \to \mathbb R$ such that $$\mathbb E|f(x) - f(y)| \asymp \sqrt{n}.$$

Indeed, for any such function, by the Gaussian concentration for Lipschitz functions (see e.g. Theorem 2.4, page 31), $$P(|f(x)-Ef(x)|\ge t)\le2e^{-t^2/2}$$ for all $t\ge0$. So,
$$E|f(x)-Ef(x)|=\int_0^\infty P(|f(x)-Ef(x)|\ge t)\,dt\le\sqrt{2\pi},$$ and hence also $E|f(y)-Ef(x)|=E|f(y)-Ef(x)|\le\sqrt{2\pi}$, so that $$E|f(x)-f(y)|\le E|f(x)-Ef(x)|+E|f(y)-Ef(x)|\le2\sqrt{2\pi}=o(\sqrt n).$$

It also follows that $$\sup_{f\in\text{Lip}(1)}E|f(x) - f(y)|\asymp1.$$