A face of the polytope of doubly stochastic matrices.

We consider a face of the polytope of doubly stochastic matrices, whose non-zero entries coincide with that of \[ V_{l,m,n}=\left[\begin{matrix} 0_{l,l} & 0_{l,m} & J_{l,n} \\ 0_{m,l} & I_{m} & J_{m,n}\\ J_{n,l} & J_{n,m} & J_{n,n}\\ \end{matrix}\right]. \] V l , m , n = [ 0 l , l 0 l , m J l , n 0 m , l I m J m , n J n , l J n , m J n , n ]. Here, $ 0_{r,s} $ 0 r , s is the $ r\times s $ r × s zero matrix, $ J_{u,v} $ J u , v denotes the $ u\times v $ u × v matrix all of whose entries are 1 and $ I_m $ I m is the identity matrix of order m. We determine the minimum permanent and minimizing matrices on this face of the polytope of doubly stochastic matrices. This research contributes towards solution of two problems from Minc's well-known lists of unsolved problems on permanents. [ABSTRACT FROM AUTHOR]