# Motivation Behind c-Convexity in Optimal Transport

I try to get into optimal transport by reading Villani's book "Optimal Transport - old and new" and I came across this definition. Unfortunately, I dont really have an intuition on what motivates them (lines (5.6)-(5.8)). They seem pretty mysterious to me.

The motivation is as follows. For the quadratic cost, i.e. $$d(x,y)^2$$, there are a bunch of basic optimal transport facts which can be proved using techniques/objects from convex analysis, including: Legendre transform, subdifferential, etc. It would be nice to be able to run those proofs for costs other than $$d(x,y)^2$$; the solution turns out to be to define a generalization of convexity relative to a given cost $$c(x,y)$$, such that, in the terminology of $$c$$-convexity, $$d(x,y)^2$$-convexity is just regular convexity.
In particular, the $$c$$-transform is the "right" generalization of the Legendre transform, the $$c$$-subdifferential is the "right" generalization of the subdifferential, etc.