The d-Majorization Polytope

We investigate geometric and topological properties of d -majorization – a generalization of classical majorization to positive weight vectors d ∈ 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 ∈ R n as an intersection of finitely many half spaces, i.e. as solutions to an inequality of the type M x ≤ 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 ≥ 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 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.