The geometry of the relay channel

Consider a memoryless relay channel, where the channel from the relay to the destination is 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 our recent work, we answered this question in the case when the channels from the source to the relay and destination are symmetric, which is the original assumption imposed by Cover, and when these channels are Gaussian. We showed that C(C
₀
) can not equal to C(∞) unless C
₀
= ∞, regardless of the SNR of the Gaussian channels, while the cut-set bound would suggest that C(∞) can be achieved at finite C
₀
. In this paper, we show that our techniques for solving Cover's problem can be naturally extended to the general Gaussian case, where the channels from the source to the relay and destination may be asymmetric, and prove an upper bound on the capacity C(C
₀
) of a general Gaussian relay channel for any C
₀
. This upper bound immediately implies that our previous conclusion, i.e. C(C
₀
) can not equal to C(∞) unless C
₀
= ∞, also holds in the asymmetric case. Our approach is geometric and relies on a strengthening of the isoperimetric inequality on the sphere by using the Riesz rearrangement inequality.