Given a cost function and two probability measures on a compact
manifold, the Monge problem is concerned with the study of minimizers of a
transportation cost among transport maps between these two measures while
the Kantorovitch problem is concerned with optimal transport plans. We
will explain the main difference between these two optimization problems.
Then we will address the uniqueness of optimal transport plans for the
Kantorovitch problem. We will show that the uniqueness issue is related
with the properties of some dynamics associted to the cost.
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