"The Capacity of the Relay Channel": Solution to Cover's Problem in the Gaussian Case

Consider a memoryless relay channel, where the relay is connected to the destination with an isolated bit pipe of capacity C0. Let C(C0) denote the capacity of this channel as a function of C0. What is the critical value of C0 such that C(C0) first equals C(∞)? This is a long-standing open problem posed by Cover and named “The Capacity of the Relay Channel,” in Open Problems in Communication and Computation, Springer-Verlag, 1987. In this paper, we answer this question in the Gaussian case and show that C(C0) can not equal to C(∞) unless C0 =∞, regardless of the SNR of the Gaussian channels. Our proof significantly deviates from the typical converse program in network information theory, which focuses on “single-letterizing” expressions involving information measures in a highdimensional space (so called n-letter forms). Instead, we develop a geometric technique that allows us to directly quantify the interplay between n-letter information measures. Our key geometric ingredient is a strengthening of the classical isoperimetric inequality on the sphere, which we develop using the Riesz rearrangement inequality. I. PROBLEM SETUP AND MAIN RESULT In 1987, Thomas M. Cover formulated a seemingly simple question in Open Problems in Communication and Computation, Springer-Verlag [2], which he called “The Capacity of the Relay Channel”. This problem, not much longer than a single page in [2], remains open to date. His problem statement, taken verbatim from [2] with only a few minor notation changes, is as follows: The Capacity of the Relay Channel Consider the following seemingly simple discrete memoryless relay channel: Here Z and Y are condi-