On the Number of Vertices of the Stochastic Tensor Polytope
Linear and Multilinear Algebra
Birkhoff polytope, Birkhoff-von Neumann theorem, Doubly stochastic matrix, Extreme point, Hypermatrix, Multidimensional matrix, Polytope, Stochastic semi-magic cube, Stochastic tensor, Vertex
This paper studies lower and upper bounds for the number of vertices of the polytope of n x n x 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 et al. [Ann Funct Anal. 2016;7(3):386–393] and also sharper than the one in Linial and Luria’s [Discrete Comput Geom. 2014;51(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.
Li, Zhongshan; Zhang, Fuzhen; and Zhang, Xiao-Dong, "On the Number of Vertices of the Stochastic Tensor Polytope" (2017). Mathematics Faculty Articles. 213.