  This paper is devoted to the study of lower and upper bounds for the number
of vertices of the polytope of $n\times n\times n$ stochastic tensors (i.e.,
triply stochastic arrays of dimension $n$). By using known results on polytopes
(i.e., the Upper and Lower Bound Theorems), we present some new lower and upper
bounds. We show that the new upper bound is tighter than the one recently
obtained by Chang, Paksoy and Zhang [Ann. Funct. Anal. 7 (2016), no.~3,
386--393] and also sharper than the one in Linial and Luria's [Discrete Comput.
Geom. 51 (2014), no.~1, 161--170]. We demonstrate that the analog of the lower
bound obtained in such a way, however, is no better than the existing ones.
