Minoration via Mixed Volumes and Cover's Problem for General Channels

TL;DR

This paper solves a long-standing open problem by providing a complete characterization of the capacity of a relay channel in a general discrete memoryless setting, using convex geometry and mixed-volume inequalities.

Contribution

It introduces a novel approach combining convex geometry and mixed-volume inequalities to analyze channel capacity without additional assumptions.

Findings

Complete solution to Cover's open problem from 1987.

New lower bounds on soft-max of stochastic processes.

Application of Minkowski's inequality to information theory.

Abstract

We give a complete solution to an open problem of Thomas Cover in 1987 about the capacity of a relay channel in the general discrete memoryless setting without any additional assumptions. The key step in our approach is to lower bound a certain soft-max of a stochastic process by convex geometry methods, which is based on two ideas: First, the soft-max is lower bounded in terms of the supremum of another process, by approximating a convex set with a polytope with bounded number of vertices. Second, using a result of Pajor, the supremum of the process is lower bounded in terms of packing numbers by means of mixed-volume inequalities (Minkowski's first inequality).