University of Toronto

mccann@math.toronto.edu

**Abstract**: Square *N* by *N matrices with non-negative
entries
whose columns and rows all sum to one form a convex set with N! vertices.
Each vertex corresponds to a permutation on N letters, according to a
theorem of Birkhoff and von Neumann.*

In 1948 Birkhoff asked what the infinite dimensional analog of this theorem
should be. We resolve this problem, by giving a condition on the support of a
measure on the square which characterizes extremality among non-negative
measures with the same *x*- and *y*-marginal projections.
Our condition is used
to derive a new sufficient condition for uniqueness of minimizer in
Kantorovich's optimal transportation problem, in terms of the topology of the
transportation cost.