  Kantorovich potentials denote the dual solutions of the renowned optimal
transportation problem. Uniqueness of these solutions is relevant from both a
theoretical and an algorithmic point of view, and has recently emerged as a
necessary condition for asymptotic results in the context of statistical and
entropic optimal transport. In this work, we challenge the common perception
that uniqueness in continuous settings is reliant on the connectedness of the
support of at least one of the involved measures, and we provide mild
sufficient conditions for uniqueness even when both measures have disconnected
support. Since our main finding builds upon the uniqueness of Kantorovich
potentials on connected components, we revisit the corresponding arguments and
provide generalizations of well-known results. Several auxiliary findings
regarding the continuity of Kantorovich potentials, for example in geodesic
spaces, are established along the way.
