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This paper is concerned with the extreme points of the polytopes ofstochastic tensors. By a tensor we mean a multi-dimensional array over the realnumber field. A line-stochastic tensor is a nonnegative tensor in which the sumof all entries on each line (i.e., one free index) is equal to 1; aplane-stochastic tensor is a nonnegative tensor in which the sum of all entrieson each plane (i.e., two free indices) is equal to 1. In enumerating extremepoints of the polytopes of line- and plane-stochastic tensors of order 3 anddimension $n$, we consider the approach by linear optimization and present newlower and upper bounds. We also study the coefficient matrices that define thepolytopes.

Enumerating extreme points of the polytopes of stochastic tensors: an optimization approac