The Muirhead-Rado inequality, 1 Vector majorization and the permutohedron

TL;DR

This paper explores the relationships between vector majorization, the Muirhead-Rado inequality, and permutohedra, establishing equivalences involving doubly stochastic matrices and geometric structures.

Contribution

It provides new proofs of classical theorems linking majorization, permutation groups, and permutohedra, clarifying their geometric and algebraic connections.

Findings

Majorization characterized by doubly stochastic matrices

Equivalence between majorization and permutohedron membership

Connections between classical inequalities and geometric structures

Abstract

Let $\mathbf{a}$ and $\mathbf{b}$ be vectors in $\mathbf{R}^n$ with nonnegative coordinates. Permuting the coordinates, we can assume that $a_1 \geq \cdots \geq a_n$ and $b_1 \geq \cdots \geq b_n$. The vector $\mathbf{a}$ majorizes the vector $\mathbf{b}$, denoted $\mathbf{b} \preceq \mathbf{a}$, if $\sum_{i=1}^n b_i = \sum_{i=1}^n a_i$ and $\sum_{i=1}^k b_i \leq \sum_{i=1}^k a_i$ for all $k \in \{1,\ldots,n-1\}$. This paper proves theorems of Hardy-Littlewood-P\'olya and Rado that $\mathbf{b} \preceq \mathbf{a}$ if and only if $P\mathbf{a} = \mathbf{b}$ for some doubly stochastic matrix $P$ if and only if $\mathbf{b}$ is in the $S_n$-permutohedron generated by $\mathbf{a}$.