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

TL;DR

This paper investigates the number of vertices of polytopes formed by polystochastic matrices, establishing that for order 3, the number of vertices grows doubly exponentially with the dimension.

Contribution

It provides new asymptotic bounds on the vertices of the polytope of polystochastic matrices, especially proving doubly exponential growth for order 3.

Findings

Number of vertices of  d polytopes is doubly exponential in dimension d.

Comparison of known bounds on vertices of -dimensional polytopes.

Establishment of asymptotic growth rates for vertices of polystochastic matrix polytopes.

Abstract

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 $\Omega_n^d$. In the present paper, we compare known bounds on the number of vertices of the polytope $\Omega_n^d$ and prove that the number of vertices of $\Omega_3^d$ is doubly exponential on $d$.