Optimal transport minimizes the cost of moving one probability measure to another. For probability measures 
and 
and a cost 
, its transport-plan formulation is
 |
where 
consists of probability measures on pairs
whose marginals are 
and 
.
The transport-map formulation instead minimizes 
over measurable
maps satisfying 
,
meaning 
for every measurable set 
. A plan permits one source point
to distribute mass among several targets; a map specifies one target for almost
every source point. The quadratic Euclidean cost gives
Brenier maps and the Wasserstein
distance. The Ma-Trudinger-Wang condition
concerns regularity for squared Riemannian distance.
