# Combinatorial Entropy Power Inequalities: A Preliminary Study of the Stam region

## TL;DR

This paper introduces the Stam region, a new mathematical construct related to entropy powers of subset sums of independent random vectors, and explores its properties and bounds.

## Contribution

It establishes that fractionally superadditive set functions bound the Stam region and proves the non-supermodularity of entropy power sums, advancing understanding of entropy inequalities.

## Key findings

- Fractionally superadditive functions bound the Stam region.
- Entropy power of sums is not supermodular in any dimension.
- The closure of the Stam region is a logarithmically convex cone.

## Abstract

We initiate the study of the Stam region, defined as the subset of the positive orthant in $\mathbb{R}^{2^n-1}$ that arises from considering entropy powers of subset sums of $n$ independent random vectors in a Euclidean space of finite dimension. We show that the class of fractionally superadditive set functions provides an outer bound to the Stam region, resolving a conjecture of A. R. Barron and the first author. On the other hand, the entropy power of a sum of independent random vectors is not supermodular in any dimension. We also develop some qualitative properties of the Stam region, showing for instance that its closure is a logarithmically convex cone.