Cover's open problem: “The Capacity of the Relay Channel”

摘要

Consider a memoryless relay channel, where the channel from the relay to the destination is 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, while the cutset bound would suggest that C(∞) can be achieved at finite C0. Our approach is geometric and relies on a strengthening of the isoperimetric inequality on the sphere by using Riesz rearrangement inequality.