Convex extensions are natural ways to extend functions defined on the extreme points of a convex set to the whole set. They rely on Choquet theory, which aims at representing points of convex sets by probability measures over their extreme points.

We focus on the case of a product of convex sets, where we find a correspondence between convex extensions and Kantorovich optimal transport problems. In particular, "classical" Kantorovich costs between probability measures are recovered as convex extensions in the simplex case, and entanglement-free quantum Wasserstein distances in the (non-simplex) case of a quantum state space. Conversely, convex extensions can themselves be recovered as suitable quotients of classical Kantorovich costs.

This correspondence provides a geometric interpretation of optimal transport and allows for transferring tools between this field and the theory of convex extensions.