  The region of entropic vectors is a convex cone that has been shown to be at
the core of many fundamental limits for problems in multiterminal data
compression, network coding, and multimedia transmission. This cone has been
shown to be non-polyhedral for four or more random variables, however its
boundary remains unknown for four or more discrete random variables. Methods
for specifying probability distributions that are in faces and on the boundary
of the convex cone are derived, then utilized to map optimized inner bounds to
the unknown part of the entropy region. The first method utilizes tools and
algorithms from abstract algebra to efficiently determine those supports for
the joint probability mass functions for four or more random variables that
can, for some appropriate set of non-zero probabilities, yield entropic vectors
in the gap between the best known inner and outer bounds. These supports are
utilized, together with numerical optimization over non-zero probabilities, to
provide inner bounds to the unknown part of the entropy region. Next,
information geometry is utilized to parameterize and study the structure of
probability distributions on these supports yielding entropic vectors in the
faces of entropy and in the unknown part of the entropy region.
