  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.
