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

Mapping the Region of Entropic Vectors with Support Enumeration & Information Geometry