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

A multidimensional nonnegative matrix is called polystochastic if the sum of entries in each line is equal to $1$. The set of all polystochastic matrices of order $n$ and dimension $d$ is a convex polytope ${\mathrm{\Omega }}_n^d$. In the present paper, we compare known bounds on the number of vertices of the polytope ${\mathrm{\Omega }}_n^d$ and prove that the number of vertices of ${\mathrm{\Omega }}_3^d$ is doubly exponential on $d$.