Entropy Functions on Two-Dimensional Faces of Polymatroid Region with One Extreme Ray Containing Rank-One Matroid

TL;DR

This work advances the entropy-region program by dissecting 2D faces of the polymatroid region $\,\Gamma_n$ that have one extreme ray containing a rank-1 matroid, and by classifying all such faces with the other extreme ray supporting a matroid into four distinct types: all-entropic, Matúš-type, Chen-Yeung-type, and non-entropic. The authors develop a framework combining matroid theory and polyhedral geometry, using p-characteristic sets and restricted faces $F_C$ to trace how entropy functions arise on each face $(M,U_{1,{n'}}^n)$. They provide complete characterizations for the cases $(M,U_{1,1}^n)$ and $(M,U_{1,2}^n)$ and extend the analysis to general $(M,U_{1,{n'}}^n)$, establishing a unified four-type classification. These results enhance understanding of the geometry of the entropy region and the boundary between entropic and non-entropic behaviors, with potential implications for information inequalities and network information theory.

Abstract

Characterization of entropy functions is of fundamental importance in information theory. By imposing constraints on their Shannon outer bound, i.e., the polymatroidal region, one obtains the faces of the region and entropy functions on them with special structures. In this paper, we characterize entropy functions on 2-dimensional faces of polymatroid region of degree n with one extreme ray containing rank-1 matroid. We classify all such 2-dimensional faces with another extreme ray containing a matroid into four types.

Figures (1)

• Figure 1: Chen-Yeung-type and Matúš-type faces

Theorems & Definitions (27)

• Definition 1
• Example 1
• Lemma 1
• Proposition 2
• proof
• Lemma 3
• Lemma 4
• Lemma 5
• Lemma 6
• proof