Calculus of Variations and Geometric Measure Theory

D. Lesesvre - P. Pegon - F. Santambrogio

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

created by santambro on 01 Oct 2012
modified by pegon on 09 Apr 2024


Published Paper

Inserted: 1 oct 2012
Last Updated: 9 apr 2024

Journal: Set-Valued and Variational Analysis
Volume: 21
Pages: 541–556
Year: 2013
Doi: 10.1007/s11228-013-0229-4

ArXiv: 1210.0761 PDF
Links: HAL repository


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 $\omega_\delta(T):=\sup_{\vert x-y\vert<\delta}\vert T(x)-T(y) \vert$. 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.

Keywords: optimal transportation, Monge-Kantorovich, modulus of continuity, monotone transports, privacy protection