Is the Wasserstein distance a convex measure?
$$W_{p}(\mu, \nu):=\left(\inf _{\pi \in \Pi(\mu, \nu)} \int_{M \times M} |x-y|^{p} \mathrm{d} \pi(x, y)\right)^{1 / p}$$
I think optimal transport, which uses $W_p(\mu, \nu)$, is a linear programming optimization problem, rather than convex optimization, if so how can it be disproven that the above is convex and is instead a different type of objective function?