Given the metric space of the probability distributions over $A=\{1,2,...,n\}$, where the metric is the the 1st or 2nd Wasserstein distance (with respect to the metric $d(i,j)=|i-j|,\ \ i,j\in A$), is it possible to equip the space with a Riemannian inner product (metric) such that the space is a complete Riemannian manifold, and the length of the shortest geodesic between any two points (probability distributions) equals the Wasserstein distance between them?
If the answer is positive:
- Is there an explicit representation of the Riemannian metric?
- Do geodesics in this space have some interpretation as the optimal mass transport paths?