A Brenier map is a quadratic-cost optimal transport map that is the gradient of a convex
function used as a potential. For an absolutely
continuous source probability measure 
on 
and a target probability
measure 
, both with finite second
moments, this map minimizes
 |
subject to 
. It is unique up to a set
of 
-measure
zero and has the form 
for a convex function 
.
