Polytopes of Stochastic Tensors

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

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