Book Inequalities

Information theoretical inequalities have strong ties with polymatroids and their representability. A polymatroid is entropic if its rank function is given by the Shannon entropy of the subsets of some discrete random variables. The book is a special iterated adhesive extension of a polymatroid with the property that entropic polymatroids have n-page book extensions over an arbitrary spine. We prove that every polymatroid has an n-page book extension over a single element and over an all-but-one-element spine. Consequently, for polymatroids on four elements, only book extensions over a two-element spine should be considered. Matúš proved that the Zhang-Yeung inequalities characterize polymatroids on four elements which have such a two-page book extension. The n-page book inequalities, defined in this paper, are conjectured to characterize polymatroids on four elements which have n-page book extensions over a two-element spine. We prove that the condition is necessary; consequently, every book inequality is an information inequality on four random variables. Using computer-aided multiobjective optimization, the sufficiency of the condition is verified up to nine-page book extensions.