On the geometry of geodesics in discrete optimal transport

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