Faces of the polytope of doubly substochastic matrices (Q2070838)

A doubly substochastic matrix is a square nonnegative matrix such that each row and each column sums at most one. The set of all \(n\times n\) doubly substochastic matrices forms a convex polytope \(\omega_{n}\) of dimension \(n^{2}\). The paper provides necessary and sufficient conditions for a face of \(\omega_{n}\) to be nonempty, and descriptions of all 1-dimensional faces, 2-dimensional faces, and facets of \(\omega_{n}\). It is worth to recall that \textit{R. A. Brualdi} and \textit{P. M. Gibson} [J. Comb. Theory, Ser. A 22, 194--230 (1977; Zbl 0355.15013)] characterised the faces of the polytope obtained from the set of all \(n\times n\) doubly stochastic matrix, square nonnegative matrices such that each row and each column sums to one.