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.

翻译：本文刻画了四阶多拟阵区域$\Gamma_4$的二维面上的熵函数。我们枚举了$\Gamma_4$所有59种类型的二维面，并完整刻画了其中27种类型上的熵函数，其中4种类型为非平凡情形。