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 
.
