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