Entropy Functions on Two-Dimensional Faces of Polymatroidal Region of Degree Four

In this paper, we characterize entropy functions on the 2-dimensional faces of the polymatroidal region $\Gamma_4$. We enumerate all 59 types of 2-dimensional faces of $\Gamma_4$ and fully characterized entropy functions on 27 types of them, among which 4 types are non-trivial.