  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.
