*Published Paper*

**Inserted:** 7 may 2008

**Journal:** SIAM Journal of Mathematical Analysis

**Volume:** 40

**Number:** 1

**Pages:** 1-20

**Year:** 2008

**Abstract:**

We consider the non-nonlinear
optimal transportation problem of minimizing the cost functional
$\C_\infty(\lambda)= \lambda\text{-}\esssup_{(x,y) \in \Omega^2}

y-x

$
in the set of probability measures on $\Omega^2$ having prescribed marginals. This corresponds to the question of characterizing the measures that realize the infinite Wasserstein distance. We establish the existence of ``local'' solutions and characterize this class with the aid of an adequate version of cyclical monotonicity. Moreover, under natural assumptions, we show that local solutions are induced by transport maps.

**Keywords:**
Monge problem, infinite Wasserstein distance, restrictable solutions, infinite cyclical monotonicity

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