Cover's open problem: “The Capacity 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<inf>0</inf>. Let C(C<inf>0</inf>) denote the capacity of this channel as a function of C<inf>0</inf>. What is the critical value of C<inf>0</inf> such that C(C<inf>0</inf>) 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<inf>0</inf>) can not equal to C(∞) unless C<inf>0</inf> = ∞, regardless of the SNR of the Gaussian channels, while the cutset bound would suggest that C(∞) can be achieved at finite C<inf>0</inf>. Our approach is geometric and relies on a strengthening of the isoperimetric inequality on the sphere by using Riesz rearrangement inequality.

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