Semiconvexity of (weak) Kantorovich potentials in the Lorentzian optimal transport problem

We study semiconvexity properties of (weak) Kantorovich potentials for the Lorentzian optimal transport problem with the standard cost function $c$.

We show that, in general, this regularity - known in the Riemannian context - does not extend to the Lorentzian setting.

Nevertheless, we provide a general regularity result for $c$-convex functions and show that, under suitable general assumptions on the measures, this yields semiconvexity of the (weak) potentials at least on an open set of full measure.

This, in turn, allows us to conclude the existence and uniqueness of an optimal transport map.