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](http://www.numdam.org/item/ASNSP_2010_5_9_2_297_0/).



Hyperbolic space embeds into Wasserstein space
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.


