The d-Majorization Polytope

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