On the number of vertices of the stochastic tensor polytope

TL;DR

This paper investigates bounds on the number of vertices of the polytope of n×n×n stochastic tensors, providing new upper bounds that improve upon recent results, but finds existing lower bounds are sufficient.

Contribution

It introduces tighter upper bounds for the vertices of the stochastic tensor polytope using known polytope theorems, improving upon recent literature.

Findings

New upper bounds are tighter than previous results.

Existing lower bounds are not improved by the new methods.

The analog of the lower bound remains comparable to existing bounds.

Abstract

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.