Geodesics and generalized geodesics

$W_{2}(\mu ,\nu ):=\min _{\gamma \in \Gamma (\mu ,\nu )}\left(\int |x_{1}-x_{2}|^{2}\,d\gamma (x_{1},x_{2})\right)^{1/2}$

Statement of Theorem

Theorem.(Benamow-Brenier) Let $\mu ,\nu \in P_{2}(R^{d})$,

<math> w_{2}^{2}(\mu,\vu)=\inf_{(\mu(t).\nu(t))}{\int_{0}^{1}} <\math>