On the vertices of the d-dimensional Birkhoff polytope

Abstract

Consider the Birkhoff polytope of n by n doubly-stochastic matrices. As the Birkhoff-von Neumann theorem famously states, its vertex set coincides with the set of all n by n permutation matrices. Here we seek a higher-dimensional analog of this basic fact. Namely, consider the polytope which consists of all tristochastic arrays of order n. These are n by n by n arrays with nonnegative entries in which every line sums to 1. What can be said about its 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, but as we show, such vertices constitute only a vanishingly small part of the total number of vertices. More concretely, we show that the number of vertices is at least (L_n)^{3/2-o(1)}, where L_n is the number of order-n Latin squares. We also briefly consider similar problems concerning the polytope of n by n by n arrays where the entries in every coordinate hyperplane sum to 1. Several open questions are presented as well.