On certain convex matrix sets (Q581622)

For any matrix \(X=(x_{ij})_{m\times n}\), \(X\geq 0\) (resp. \(X>0)\) means that all \(x_{ij}\geq 0\) (resp. all \(x_{ij}>0)\). For \(r=(r_ 1,...,r_ m)^ T\geq 0\), \(c=(c_ 1,...,c_ n)^ T\geq 0\), \(r^ Tu_ m=c^ Tu_ n=s\), where each \(u_ k\) is a column of k ones, \(U(r,c)=\{(A)_{m\times n}\geq 0:Au_ n=r,A^ Tu_ m=c\}\), \(U^+(r,c)=\{(A)_{m\times n}+(P)_{m\times n}:\) \(A\in U(r,c),P\geq 0\}\), and \(U(\geq r,\geq c)=\{(A)_{m\times n}\geq 0:\) \(Au_ n-r\geq 0\), \(A^ Tu_ m-c\geq 0\}\). It is shown that the extreme points of each of U(r,c) and \(U^+(r,c)\) are the extreme points of U(\(\geq r,\geq c)\) which satisfy \(u_ m^ TAu_ n=s\).