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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.
  • $\mathcal{Z} = (\operatorname{Lip}([0,1]^2))^*$ 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 $(\operatorname{Lip}_0(X))^*$ embeds into $\mathcal{P}_1(X)$ with distortion arbitrarily close to 1. (Here $\operatorname{Lip}_0(X)$ 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 $(\operatorname{Lip}_0(\mathcal{N}))^*$, where $\mathcal{N}$ is a small net in $[0,1]^2$. But I can't figure it out or locate an argument in [71].

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What the reference is saying is: for every finite metric space $X$, and every $D>0$, there is an embedding $f\colon (\left(\mathrm{Lip}_0(X)\right)^*\to \mathcal{P}_1(X)$ of distortion at most $D$.

Now suppose $X\subset \cal{Z}$ and observe that this induces an isometric embedding $\mathcal{P}_1(X) \to \mathcal{P}_1(\mathcal{Z})$ (every measure on $X$ is a convex combination of delta-measures, which make sense as measures on $\mathcal{Z}$ as well). In particular the low-distortion map $f$ above is also a low-distortion map into $\mathcal{P}_1(\mathcal{Z})$.

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  • $\begingroup$ $\mathcal{P}_1(\mathcal{Z})$ is a monster space $\endgroup$ Commented Feb 16, 2021 at 23:15
  • $\begingroup$ The target space is much more modest in the claim. $\endgroup$ Commented Feb 16, 2021 at 23:21
  • $\begingroup$ Hey Lior, thanks for thinking about the thing and replying. $\endgroup$ Commented Feb 16, 2021 at 23:29
  • $\begingroup$ Sorry -- my answer is totally wrong. I'll try fixing it. $\endgroup$ Commented Feb 17, 2021 at 0:21

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