On the Vertices of the d-Dimensional Birkhoff Polytope

Let us denote by Ωn the Birkhoff polytope of n× n doubly stochastic matrices. As the Birkhoff–von Neumann theorem famously states, the vertex set of Ωn coincides with the set of all n× n permutation matrices. Here we consider a higherdimensional analog of this basic fact. Let Ω n be the polytope which consists of all tristochastic arrays of order n. These are n×n×n arrays with nonnegative entries in which every line sums to 1. What can be said about Ω n ’s vertex set? It is well known that an order-n Latin square may be viewed as a tristochastic array where every line contains n − 1 zeros and a single 1 entry. Indeed, every Latin square of order n is a vertex of Ω n , but as we show, such vertices constitute only a vanishingly small subset of Ω n ’s vertex set. More concretely, we show that the number of vertices of Ω (2) n is at least (Ln) 3 2−o(1), where Ln is the number of order-n Latin squares. We also briefly consider similar problems concerning the polytope of n× n× n arrays where the entries in every coordinate hyperplane sum to 1, improving a result from Kravtsov (Cybern. Syst. Anal., 43(1):25–33, 2007). Several open questions are presented as well.