The \(L^\infty\)-optimal transport problem is studied, namely the problem \[ \min \gamma - \mathrm{ess}\sup c(x,y) \] where \(c\) is the cost function and the minimum is taken among all the transport plans \(\gamma\) between two given probability measures \(\mu\) and \(\nu\) (in this paper \(\mu\) and \(\nu\) typically live in \(\mathbb{R}^d\)).
The authors introduce a kind of twist condition for the cost \(c\), suitable for \(L^\infty\)-problems, that together with some other technical assumptions allow to infer that a \(\infty\)-monotone transport plan is induced by a transportation map (the precise statements are given in Theorem 2.17 and Theorem 2.19). Under the same assumptions for the cost \(c\), another result of the paper (Theorem 3.3) shows the uniqueness of \(\infty\)-\(c\)-cyclically Monge minimizers of problems where \(\mu\) is absolutely continuous with respect to the Lebesgue measure.
In the final part of the paper there are some examples of costs that satisfy the assumptions of the results.