Weak k-Majorization and Polyhedra

For integers k and n with k ≤ n a vector x ∈ ℝⁿ is said to be weakly k-majorized by a vector q ∈ ℝ^k if the sum of the r largest components of x does not exceed the sum of the r largest components of q, for r = 1,⋯,k. For a given q the set of vectors weakly k-majorized by q defines a polyhedron P(q; k). We determine the vertices of both P(q; k) and its integer hull Q(q; k). Furthermore a complete and nonredundant linear description of Q(q; k) is given