Thanks for everyone's help with understanding finite metric embeddings in Euclidean space. I have a follow-up question. Say we have the Wasserstein distance between $n$ distributions in Euclidean space, i.e. each distribution's support is $\mathbb{R}^k$ and the cost is the Euclidean distance in $\mathbb{R}^k$. This set of distances is a finite metric. Can this metric be embedded in a Euclidean space, i.e. can each distribution be mapped to a point in a Euclidean space so all pairwise distances between them are the same as the Wasserstein distance between the corresponding distributions?
I understand this is not possible for Wasserstein distance with arbitrary cost, since not all finite metrics can be embedded in Euclidean space. But this may be possible if the transport cost is the Euclidean distance in the distribution support?
