Faces of the polytope of doubly substochastic matrices

Abstract

We study the faces of the convex polytope of all n×n doubly substochastic matrices, denoted by ωn. We give the necessary and sufficient conditions of a face being nonempty. We also describe all 1-dimensional faces, 2-dimensional faces, and facets of ωn. Moreover, we explore the relation between the faces of ωn and the faces of Ωn, the convex polytope of all n×n doubly stochastic matrices.