Asymptotic bounds on the numbers of vertices of polytopes of polystochastic matrices

A multidimensional nonnegative matrix is called polystochastic if the sum of the entries in each line is equal to 1. The set of all polystochastic matrices of order n and dimension d forms a convex polytope Ω n d . In the present paper, we compare known bounds on the number of vertices of the polytope Ω n d and prove that the number of vertices of Ω 3 d is doubly exponential in d .