$L^\infty$-optimal transport for a class of quasiconvex cost functions

We consider the $L^\infty$-optimal mass transportation problem $\min_{Π(μ, ν)} γ-\mathrm{ess\,sup\,} c(x,y),$ 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.

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