Diagonal sums of doubly substochastic matrices

Let Ω denote the convex polytope of all n × n doubly stochastic matrices, and ω denote the convex polytope of all n × n doubly substochastic matrices. For a matrix A ∈ ω , define the sub-defect of A to be the smallest integer k such that there exists an (n + k) × (n + k) doubly stochastic matrix containing A as a submatrix. Let ω denote the subset of ωn which contains all doubly substochastic matrices with sub-defect k. For π a permutation of symmetric group of degree n, the sequence of elements a , a , …, a is called the diagonal of A corresponding to π. Let h(A) and l(A) denote the maximum and minimum diagonal sums of A ∈ ω , respectively. In this paper, existing results of h and l functions are extended from Ω to ω . In addition, an analogue of Sylvesters law of the h function on ω is proved.