Characterization of a class of weak transport-entropy inequalities on the line

Understand

We study an optimal weak transport cost related to the notion of convex order between probability measures.

• On the real line, we show that this weak transport cost is reached for a coupling that does not depend on the underlying cost function.
• As an application, we give a necessary and sufficient condition for weak transport-entropy inequalities in dimension one.
• In particular, we obtain a weak transport-entropy form of the convex Poincar{\'e} inequality in dimension one.