A Birkhoff Theorem for Doubly Stochastic Matrices with Vector Entries

The concept of a doubly stochastic matrix whose entries come from a convex subset of the unit square is defined. It is proved that the only convex subsets of the unit square which contain (0,0) and (1, 1) and allow an extension of Birkhoff's characterization of the extreme points of the set of doubly stochastic matrices are parallelograms. A sufficient condition is given for a matrix to be extreme when the convex subset is not a parallelogram.