The extreme points of centrosymmetric transportation polytopes

Denote by Uπ(R,S) the convex set of nonnegative centrosymmetric matrices with given row sum vector R and column sum vector S, and denote by U≤π(R,S) the convex set of nonnegative centrosymmetric matrices with the row sum vector componentwisely dominated by R and the column sum vector componentwisely dominated by S respectively. We characterize all extreme points of Uπ(R,S) and U≤π(R,S). In addition, we show that the extreme points of Ωnπ, the polytope of all n×n centrosymmetric doubly stochastic matrices, and the extreme points of ωnπ, the polytope of all n×n centrosymmetric doubly substochastic matrices, can be obtained by letting R=S=(1,1,…,1) in Uπ(R,S) and U≤π(R,S) respectively.