# Hyperbolic space embeds into Wasserstein space

Fix a positive integer $$n$$, let $$\mathbb{H}^n$$ be the $$n$$-dimensional hyperbolic space, $$r>0$$, $$x\in \mathbb{H}^n$$ and consider the closed (compact) geodesic ball $$B_{\mathbb{H}^n}(x,r)$$. Are there known estimates on the minimum distortion of a bi-Lipschitz embedding of $$B_{\mathbb{H}^n}(x,r)$$ into $$\mathcal{W}_2(\mathbb{R})$$.

Let me just note that a bi-Lipschitz embedding into $$\mathcal{W}_2(\mathbb{R})$$ must exist by this paper.

For global distortion.

Choose a 4 point set --- vertices of an equilateral triangle in $$B(x,r)_{\mathbb{H}^n}$$ and its center and observe that it cannot be emebdded isometrically in nonnegatively curved space in the sense of Alexandrov, which includes $$\mathcal{W}_2(\mathbb{R})$$.

In is straightforward to get some explicit bounds this way, most likely they are far from being optimal.

• Just apply the (3+1)-point comparison, you will get that for any $\varepsilon>0$ and all sufficiently large $R$ the map cannot be $[\sqrt3,2-\varepsilon]$-bilipschitz. Commented Apr 3, 2022 at 16:48
• By the way, if you apply (n+1)-point comparison you could get that distortion should be at least $\sqrt2-\varepsilon$ for large $R$. Most likely the bound should approach infinity, but I do not see an argument to prove it. Commented Apr 3, 2022 at 16:55
• May I ask you to add the details for this, I'm really not an expert in this type of argument :/ Commented Apr 8, 2022 at 17:33
• @Carlos_Petterson, I guess it is about (3+1)-point comparison --- check our book arxiv.org/abs/1903.08539 Commented Apr 8, 2022 at 18:08

$$\mathcal{W}_2(\mathbb{R})$$ is isometric to a convex subset of a Hilbert space (and embedding a hyperbolic space into Euclidean/Hilbert one has a lot of history).

The argument can find an argument in S.S. Vallender "Calculation of the Wasserstein distance between probability distributions on the line". I will do an informal sketch.

I want to see probability measures in $$\mathbb{R}$$ as piles of sand with sand grains enumerated from left to right and indexed by $$[0,1]$$. For a measure $$\mu$$ define function $$\Phi_{\mu}:[0,1]\rightarrow \mathbb{R}$$ by $$\Phi_\mu(t)$$ be a position of $$t$$-th grain in $$\mathbb{R}$$. I claim that for every $$\mu, \nu$$ $$\mathcal{W}_2(\mu,\nu) = ||\Phi(\mu)-\Phi(\nu)||_{L_2}.$$

Before proving this let say how the optimal transport looks on $$\mathbb{R}$$.

Lemma 1: for the measures $$\mu$$ and $$\nu$$ the optimal transport plan between them sends $$t$$-th sand grain of $$\mu$$ into $$t$$-th sand grain of $$\nu$$.

I give the proof of the lemma for the case when $$\mu = \frac{1}{n}(\delta(\mu_1) + \dots + \delta(\mu_n)),$$ for some real $$\mu_1 < \dots < \mu_n$$ and $$\nu = \frac{1}{n}(\delta(\nu_1) + \dots + \delta(\nu_n))$$, for some real $$\nu_1 < \dots < \nu_n$$. I need to show that optimal transport moves $$\frac{1}{n} \delta(\mu_1)$$ into $$\frac{1}{n}\delta(\nu_1)$$ and then the rest will follow by induction. Okay, lets assume its not and (a part of) $$\frac{1}{n} \delta(\mu_1)$$ is moved to $$\frac{1}{n} \delta(\nu_k)$$ and (a part of) $$\frac{1}{n}\delta(\mu_j)$$ is moved to $$\frac{1}{n} \delta(\nu_1)$$ for some $$j,k > 1$$. I claim that if we change our transport by moving (the part of) $$\frac{1}{n} \delta(\mu_1)$$ into $$\frac{1}{n}\delta(\nu_1)$$ and (the part of) $$\frac{1}{n}\delta(\mu_j)$$ into $$\frac{1}{n} \delta(\nu_k)$$ it will get cheaper. Which follows from the following trivial lemma.

Lemma 2: Suppose that $$a,b,c > 0$$ then $$a^2 + (b + c)^2 < (a + c)^2 + b^2$$ iff $$b < a$$.

Now it's really easy to proof our main statement $$\mathcal{W}_2(\mu,\nu) = ||\Phi(\mu)-\Phi(\nu)||_{L_2}.$$
for the case when $$\mu = \frac{1}{n}(\delta(\mu_1) + \dots + \delta(\mu_n)),$$ for some real $$\mu_1 < \dots < \mu_n$$ and $$\nu = \frac{1}{n}(\delta(\nu_1) + \dots + \delta(\nu_n))$$, for some real $$\nu_1 < \dots < \nu_n$$. Indeed, squares of both sides are equal to $$\frac{1}{n}\sum_{i=1}^{n}|\mu_i - \nu_i|^2.$$

PS: note that measures of the type $$\mu = \frac{1}{n}(\delta(\mu_1) + \dots + \delta(\mu_n)),$$ for some real $$\mu_1 < \dots < \mu_n$$ are dense in Wasserstein space see Proposition 2.10 from Karl-Theodor Sturm. "On the geometry of metric measure spaces.". So all the above can be formalized with some suffering.

The Nash--Kuiper theorem provides an isometric embedding of $$\mathbb{H}^n\to \mathbb{R}^{n+1}$$; moreover, its image might be in a bounded set. Now observe that $$\mathcal{W}_2(\mathbb{R})$$ contains a subset isometric to a ball in $$\mathbb{R}^{n+1}$$.
• The Nash-Kuiper theorem provides an isometric embedding (in the Riemannian sense) and not in the metric sense (as YCor pointed out to me a couple of days ago). So then $f:\mathbb{H}^n \rightarrow \mathbb{R}^{2n+1}$ is only locally Lipschitz with constant $\max_{x\in ...}\,\|\nabla f(x)\|\neq 1$. Commented Apr 2, 2022 at 12:15