Given a cost function $c:\mathbb{R}^d\times \mathbb{R}^d\rightarrow \mathbb{R}^+$ we say a function $\phi:\mathbb{R}^d\rightarrow \mathbb{R}$ is c-concave when there exists some $\psi:\mathbb{R}^d\rightarrow \mathbb{R}\cup\{-\infty\}$, $\psi\not\equiv -\infty$ such that $$ \phi(x) = \inf_{y\in\mathbb{R}^d} [c(x,y)-\psi(y)]. $$
If the cost function is the usual euclidean metric $c(x,y) = \frac{1}{2}|x-y|^2$ then we know a function $\phi(x)$ is c-concave iff $$ \frac{x^2}{2}-\phi(x) \quad \text{is convex.} $$
I was wondering whether there was some way to characterize the convexity/concavity of a c-concave function regarding the concavity/convexity of the cost. In particular I am specially interested in the case in which $c(x-y)$ is a concave function (and a distance indeed).
Thanks in advance!