  We consider the $L^\infty$-optimal mass transportation problem \[
\min_{\Pi(\mu, \nu)} \gamma-\mathrm{ess\,sup\,} c(x,y), \] for a new class of
costs $c(x,y)$ for which we introduce a tentative notion of twist condition. In
particular we study the conditions under which the infinitely-motonone
minimizers are induced by a transportation map. We also state a uniqueness
result for infinitely cyclically monotone Monge minimizers that corresponds to
this class of cost functions. We compare the results to previous works.
