Talk:Optimal Transport and the Monge Ampère equation

Beginning

• Cite specific pages where Santambrogio discusses Monge Ampere, since this is only a small part of that chapter
• The explanation of the relationship to optimal transport is a little confusing. It might be better to explain what a transport map is (perhaps linking to the page on the Monge Problem on the OT wiki) and say something along the lines of ``under sufficient regularity assumptions on the measures mu and nu, the condition that a transport map pushes forward mu to nu can be equivalently formulated in terms of the transport map solving a type of Monge-Ampere equation``
• The only time that we need the cost to be quadratic is to reduce to the case that the transport map is given by the gradient of a convex function. Maybe it's best to save explaining this technicality to the next section.

Deriving the Monge Ampere Equation

• Emphasize that the infimum is over all transport maps pushing forward mu to nu.
• There is a type-o -- an extra > sign.
• ``tell us that T pushes forward mu to nu if and only if``
• ``The above equation is a type of Monge-Ampere equation``

Notable Properties of the Monge-Ampere equation

• Move the discussion of the boundary conditions to the previous section, and start with ``Properties of solutions to the Monge Ampere equation give us information about the optimal transport map``
• Cite Villani for the theorem you quote and mention Caffarelli's contribution
• Cite Villani for the notion of Brenier solutions