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

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 $\mathbf{a}$.