Exploring the entropic region

Abstract: The paper explores three known methods, their variants and limitations, that can be used to obtain new entropy inequalities. The Copy Lemma was distilled from the original Zhang-Yeung construction which produced the first non-Shannon inequality. Its iterated version, effects of symmetrizations, and connections with polyhedral vertex enumeration are discussed. Another method, derived from the principle of maximum entropy, has the Copy Lemma as a special case. Nevertheless, none of the two presented variants is known to generate more inequalities than the iterated Copy Lemma. Finally, the Ahlswede-K\"orner method is shown to employ a hidden application of the Copy Lemma - the underlying lemma alone cannot generate new inequalities -, which makes this method strictly weaker than the Copy Lemma. The paper is written in a tutorial style and concludes with a list of open questions and research problems.