The Optimal Transport problem in $L^\infty$, i.e. the problem of minimizing the $L^\infty$-norm of the cost function among the transport plans, is a nonconvex and thus a presumably more difficult problem. Due to the success of entropic approximation and of Sinkhorn's algorithm, seeking an analogue for the infinity case seems quite natural. I will show the $\Gamma$-convergence of the regularized functionals to the one related to the OT problem in $L^\infty$ and that every cluster point of the minimizers is a $\infty$-cyclically monotone transport plan which is for some cost functions a solution of the Monge problem and thus a map.
