Optimal transportation with an oscillation-type cost: the one-dimensional case

Abstract

The main result of this paper is the existence of an optimal transport map $T$ between two given measures $\mu$ and $\nu$, for a cost which considers the maximal oscillation of $T$ at scale $\delta$, given by ${\omega }_{\delta }(T):={\sup }_{|x-y|<\delta }|T(x)-T(y)|$. 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.

References in corpus (1)

• A sharp inequality for transport maps in W^{1,p}(R) via approximation

Cited by in corpus (1)

• The Monge problem with vanishing gradient penalization: Vortices and asymptotic profile