Optimal Transportation with an Oscillation-Type Cost: The One-Dimensional Case

The main result of this paper is the existence of an optimal transport map T between two given measures μ and ν, for a cost which considers the maximal oscillation of T at scale δ, given by ω _δ (T) :?=??sup_|x???y|?<?δ |T(x)???T(y)|. The minimization of this criterion finds applications in the field of privacy-respectful data transmission. The existence proof unfortunately only works in dimension one and is based on some monotonicity considerations.