On Characterization of Entropic Vectors at the Boundary of Almost Entropic Cones

The entropy region is a fundamental object in information theory. An outer bound for the entropy region is defined by a minimal set of Shannon-type inequalities called elemental inequalities also referred to as the Shannon region. This paper focuses on characterization of the entropic points at the boundary of the Shannon region for three random variables. The proper faces of the Shannon region form its boundary. We give new outer bounds for the entropy region in certain faces and show by explicit construction of distributions that the existing inner bounds for the entropy region in certain faces are not tight.