The Muirhead-Rado inequality, 1 Vector majorization and the permutohedron
Melvyn B. Nathanson
TL;DR
The paper establishes that for vectors with nonnegative coordinates, majorization $b riangleleft a$ is equivalent to the existence of a doubly stochastic matrix $P$ with $b=Pa$, and that $b$ lies in the permutohedron $K(a)$. It develops the Hardy-Littlewood-Pólya framework of $T$-transformations, showing any majorization step can be achieved by a finite product of 2×2 doubly stochastic maps, and connects these results to the Birkhoff-von Neumann theorem, yielding a geometric interpretation via $K(a)$ and its convex combinations of permutation images. It also proves a key convexity inequality: for convex $f$, $ extstyle extstyleig( ext{sum}_i f(b_i)ig)\leig( ext{sum}_i f(a_i)ig)$. Together, these results unify majorization, doubly stochastic maps, and permutohedra, and provide constructive methods for realizing majorization via matrix transformations.
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-Pólya 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 $\mathbf{a}$.