  The optimal transport problem studies how to transport one measure to another
in the most cost-effective way and has wide range of applications from
economics to machine learning. In this paper, we introduce and study an
information constrained variation of this problem. Our study yields a
strengthening and generalization of Talagrand's celebrated transportation cost
inequality. Following Marton's approach, we show that the new transportation
cost inequality can be used to recover old and new concentration of measure
results. Finally, we provide an application of this new inequality to network
information theory. We show that it can be used to recover almost immediately a
recent solution to a long-standing open problem posed by Cover regarding the
capacity of the relay channel.
