Suppose $\Omega$ is a suitably regular domain in $\mathbb{R}^n$ and $\rho_0,\rho_1\in\textrm{Prob}(\Omega)$. Benamou and Brenier showed that the $L_2$ transportation distance between $\rho_0$ and $\rho_1$ can be computed as follows:

$$d(\rho_0,\rho_1)^2=\inf_{\rho(x,t),v(x,t)}\int_{\Omega}\int_0^1\rho(x,t)|v(x,t)|^2\,dt\,dx$$

where $\rho(x,t)$ satisfies $\rho(x,0)=\rho_0(x)$, $\rho(x,1)=\rho_1(x)$, and $\frac{\partial\rho}{\partial t}+\nabla\cdot(\rho v)=0$. This formula appears in many discretizations of optimal transportation.

We know $d(\cdot,\cdot)$ satisfies the triangle inequality because it was derived from the Wasserstein distance. But, this evidence for the triangle inequality is "heavy" in that if you *define* Wasserstein distances using the above PDE formulation, you could only prove triangle inequality after showing a (complicated) relationship with the theory of optimal transportation.

**Is it possible to prove the triangle inequality directly from the "PDE-constrained optimization" formulation written above?** Even an informal proof that ignores regularity issues etc. would be useful for me.

Thanks!

PS -- The Benamou/Brenier paper is "A computational ﬂuid mechanics solution to the Monge-Kantorovich mass transfer problem," which lives here: http://link.springer.com/article/10.1007%2Fs002110050002

PPS -- Also posted here, but I realized it's probably not the right place: https://math.stackexchange.com/posts/796634/edit