Is there a meaning of distances for 0<p<1 for Wasserstein distance?

Question

The wikipedia link for Wasserstein metric is defined for $p\in[1,\infty)$. https://en.wikipedia.org/wiki/Wasserstein_metric Given some data the distance can be calculated using an optimization routine. In the routine one can choose any value for $0<p<1$ and obtain a solution. The question is can we interpet the solution as a valid distance, with proper meaning?

Answer 1

Note that for $0<p<1$, $d_p(x,y)\equiv (d(x,y))^{p}$ is a metric on your original space. Thus, you can define a Wasserstein $p$ metric for $(d(x,y))^{p}$ as the Wasserstein $1$ metric for $d_p(x,y)$ in the usual setting.