The $d$-Majorization Polytope

TL;DR

This paper explores the geometric structure of $d$-majorization, providing a simplified halfspace description of the associated polytopes, analyzing their extreme points, and examining the induced preorder structure.

Contribution

It introduces a new characterization of $d$-majorization, derives a halfspace description of $d$-majorization polytopes, and analyzes their extreme points and order properties.

Findings

Halfspace description of $d$-majorization polytopes derived

Continuity of the $d$-majorization polytope established

Characterization of extreme points and order structure provided

Abstract

We investigate geometric and topological properties of $d$-majorization -- a generalization of classical majorization to positive weight vectors $d \in \mathbb{R}^n$. In particular, we derive a new, simplified characterization of $d$-majorization which allows us to work out a halfspace description of the corresponding $d$-majorization polytopes. That is, we write the set of all vectors which are $d$-majorized by some given vector $y \in \mathbb{R}^n$ as an intersection of finitely many half spaces, i.e. as solutions to an inequality of the type $Mx\leq b$. Here $b$ depends on $y$ while $M$ can be chosen independently of $y$. This description lets us prove continuity of the $d$-majorization polytope (jointly with respect to $d$ and $y$) and, furthermore, lets us fully characterize its extreme points. Interestingly, for $y\geq 0$ one of these extreme points classically majorizes every other element of the $d$-majorization polytope. Moreover, we show that the induced preorder structure on $\mathbb{R}^n$ admits minimal and maximal elements. While the former are always unique the latter are unique if and only if they correspond to the unique minimal entry of the $d$-vector.