The Monge-Kantorovich problem for the infinite Wasserstein distance presents
several peculiarities. Among them the lack of convexity and then of a direct
duality. We study in dimension 1 the dual problem introduced by Barron, Bocea
and Jensen. We construct a couple of Kantorovich potentials which is "as less
trivial as possible". More precisely, we build a potential which is non
constant around any point that the plan which is locally optimal moves at
maximal distance. As an application, we show that the set of points which are
displaced to maximal distance by a locally optimal transport plan is minimal
