Polytopes of stochastic tensors

Abstract

Considering $n\times n\times n$ stochastic tensors $\left(a_{ijk}\right)$ (i.e., nonnegative hypermatrices in which every sum over one index $i$, $j$, or $k$, is 1), we study the polytope (${\Omega }_n$) of all these tensors, the convex set ($L_n$) of all tensors in ${\Omega }_n$ with some positive diagonals, and the polytope (${\Delta }_n$) generated by the permutation tensors. We show that $L_n$ is almost the same as ${\Omega }_n$ except for some boundary points. We also present an upper bound for the number of vertices of ${\Omega }_n$.