# Why $(Lip([0,1]^2))^*$ is finitely representable in 1-Wasserstein space over the plane?

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

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

From  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 .

## 1 Answer

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 $$\mathcal{P}_1(\mathcal{Z})$$.

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