In "SNOWFLAKE UNIVERSALITY OF WASSERSTEIN SPACES" by ALEXANDR ANDONI, ASSAF NAOR, AND OFER NEIMAN they have the following notation:

- For a metric space X they write $\mathcal{P}_1(X)$ to denote $1$-Wasserstein space over $X$ aka Earthmover space aka Kantorovich-Rubinstein space etc.
- $\mathcal{Z} = (Lip([0,1]^2))^*$ is a dual of real-valued Lipschitz functions on the square $[0,1]^2$. (I assume that we always take the Lipschitz constant as the norm on Lipschitz functions.)

And they mention the following thing that I fail to understand:

As explained in [71], every finite subset of $\mathcal{Z}$ embeds into $\mathcal{P}_1(\mathbb{R}^2)$ with distortion arbitrarily close to 1,...

[71] A. Naor and G. Schechtman. Planar earthmover is not in L1. SIAM J. Comput., 37(3):804–826 (electronic), 2007.

From [71] I learned that by applying Kantorovich duality we can get that if $X$ is finite then every finite subset of $(Lip_0(X))^*$ embeds into $\mathcal{P}_1(X)$ with distortion arbitrarily close to 1. (Here $Lip_0(X)$ is the space of Lipschitz functions which are $0$ at some fixed point $x_0$.)

It feels like the argument should be that for a given finite subset of $\mathcal{Z}$ and a fixed D > 1 we are able to embed it with distortion < D into $(Lip_0(\mathcal{N}))^*$, where $\mathcal{N}$ is a small net in $[0,1]^2$. But I can't figure it out or locate an argument in [71].