The Entropy Region is not Closed Under Duality

TL;DR

This paper demonstrates that for five or more variables, the entropy region is not closed under a duality operation derived from polymatroid theory, resolving an open question in information theory.

Contribution

It introduces a duality concept for information inequalities and proves that the entropy region is not closed under this duality for n≥5, answering a longstanding open problem.

Findings

Entropy region not closed under duality for n≥5

Introduces duality notion from polymatroids to information inequalities

Resolves an open question from 1992

Abstract

We import a duality notion coming from polymatroids to define duality for information inequalities. We show that the entropy region for $n\ge 5$ is not closed under duality. Our result answers an open question of Mat\`u\v{s} (1992).