Nonnegative matrices with prescribed row and column sums

Let r[lowered 1],...,r[lowered m],c[lowered 1],...,c[lowered n] be a given set of positive numbers such that [the summation of r[lowered i] as i goes from 1 to m] = [the summation of c[lowered j] as j goes from 1 to n]. If [cursive A] is the class of nonnegative MXN matrices A of a given pattern, certain conditions are defined which determine whether the set of positive numbers is consistent for the pattern of A. Let [average of cursive A] be the set of nonnegative MXN matrices A having r[lowered i], c[lowered j] as row and column sums respectively. The set [average of cursive A] is shown to be convex. The result of the condition for consistency is used to classify the extreme points of the set [average of cursive A].