On the geometry of geodesics in discrete optimal transport

We consider the space of probability measures on a discrete set [Formula: see text] , endowed with a dynamical optimal transport metric. Given two probability measures supported in a subset [Formula: see text] , it is natural to ask whether they can be connected by a constant speed geodesic with support in [Formula: see text] at all times. Our main result answers this question affirmatively, under a suitable geometric condition on [Formula: see text] introduced in this paper. The proof relies on an extension result for subsolutions to discrete Hamilton–Jacobi equations, which is of independent interest.