Converse bounds via coupling for interference channels and proof of Costa’s conjecture

It is shown that under suitable regularity conditions, differential entropy is O( √ n)-Lipschitz as a function of probability distributions on R with respect to the quadratic Wasserstein distance. Under similar conditions, (discrete) Shannon entropy is shown to be O(n)-Lipschitz in distributions over the product space with respect to Ornstein’s d̄-distance (Wasserstein distance corresponding to the Hamming distance). These results together with Talagrand’s and Marton’s transportationinformation inequalities allow one to replace the unknown multi-user interference with its i.i.d. approximations. As an application, a new outer bound for the two-user Gaussian interference channel is proved, which, in particular, settles the “missing corner point” problem of Costa (1985).