  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.
