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

## TL;DR

This paper solves a long-standing open problem by showing that in Gaussian relay channels, the capacity cannot be achieved with a finite relay link capacity, using a new high-dimensional geometric approach.

## Contribution

The paper introduces a novel geometric method to bound the capacity of Gaussian relay channels, providing a definitive answer to Cover's problem.

## Key findings

- Capacity cannot be achieved with finite relay link capacity in Gaussian channels.
- Develops a new high-dimensional isoperimetric inequality extension.
- Provides a new upper bound on the relay channel capacity.

## Abstract

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 $C(\infty)$? 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 $C(C_0)$ can not equal to $C(\infty)$ unless $C_0=\infty$, 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.