“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 C
₀
. Let C(C
₀
) denote the capacity of this channel as a function of C
₀
. What is the critical value of C
₀
, such that C(C
₀
) 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(C
₀
) cannot equal to C(∞) unless C
₀
= ∞, regardless of the SNR of the Gaussian channels. This result follows as a corollary to a new upper bound we develop on the capacity of this channel. Instead of “single-letterizing” expressions involving information measures in a high-dimensional space as is typically done in converse results in information theory, our proof directly quantifies the tension between the pertinent n-letter forms. This is done by translating the information tension problem to a problem in high-dimensional geometry. As an intermediate result, we develop an extension of the classical isoperimetric inequality on a high-dimensional sphere, which can be of interest in its own right.