I have a question about wasserstein distance

Question

Given the wasserstein distance of order two, $$d(\mu_{1},\mu_{2})^{2}=inf_{p\in P(\mu_{1},\mu_{2})}\int_{R^{n}\times R^{n}}|x-y|^{2}p(dxdy)$$ Is there $d(\mu_{1},\mu_{2})=d(\mu_{2},\mu_{1})$ or $d(\mu_{1},\mu_{2})=-d(\mu_{2},\mu_{1})$?

Answer 1

One can show that the Wasserstein is a distance/metric on the space of probability measures, therefore we have symmetry, i.e. $d(\mu_1, \mu_2) = d(\mu_2, \mu_1)$. See for instance this reference.