The entropy region is constructed from vectors of random variables by collecting Shannon entropies of all subvectors. Its shape is studied here by means of polymatroidal constructions, notably by convolution. The closure of the region is decomposed into the direct sum of tight and modular parts, reducing the study to the tight part. The relative interior of the reduction belongs to the entropy region. Behavior of the decomposition under self-adhesivity is clarified. Results are specialized and extended to the region constructed from four tuples of random variables. This and computer experiments help to visualize approximations of a symmetrized part of the entropy region. The four-atom conjecture on the minimal Ingleton score is refuted.

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.

The information rate for an access structure is the reciprocal of the load of the optimal secret sharing scheme for this structure. We determine this value for all trees: it is (2-1/c)-1, wherecis the size of the largest core of the tree. A subset of the vertices of a tree is a core if it induces a connected subgraph and for each vertex in the subset one finds a neighbor outside the subset. Our result follows from a lower and an upper bound on the information rate that applies for any graph and happen to coincide for trees because of a correspondence between the size of the largest core and a quantity related to a fractional cover of the tree with stars.