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

TL;DR

This paper characterizes entropy functions on specific 2D faces of the polymatroid region, focusing on those with an extreme ray containing a rank-1 matroid, and classifies these faces into four types.

Contribution

It provides a complete classification of 2D faces of the polymatroid region with a rank-1 matroid extreme ray, enhancing understanding of entropy function structures.

Findings

Classification of all such 2D faces into four types

Characterization of entropy functions on these faces

Insights into the structure of polymatroid regions

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.