Are convex functions on manifolds the same as $c$-convex functions, where $c(x,y)=d(x,y)^2/2$?

Question

I am reading the following book on optimal transport. While reading I came across the following definition of $c-$convexity.

Given $X$ and $Y$ metric spaces, $c: X \times Y \rightarrow \mathbb{R}$, and $\varphi: X \rightarrow \mathbb{R} \cup$ $\{+\infty\}$, we say that $\varphi$ is c-convex if $$ \varphi(x)=\sup _{y \in Y}\left\{-c(x, y)+\lambda_y\right\} $$ for some $\left\{\lambda_y\right\}_{y \in Y} \subset \mathbb{R} \cup\{-\infty\}$

I know that when $X=Y=\mathbb{R}^d$ with $c(x,y)=-x\cdot y$ then $c-$convex functions are the usual convex functions on Euclidean space. Is the same analogy holds true when $X=Y=M$ is a Riemannian manifold and $c(x,y)=d(x,y)^2/2$? By convex functions on manifolds I mean functions that are geodesically convex.

Answer 1

Note that the function $f\colon x\mapsto -c(x_0,x)$ is $c$-convex. Further for $c(x,y)=|x-y|^2_g/2$, the function $f$ is not convex in a neighborhood of $x_0$. So the answer is "no".

However, if the manifold has nonnegative sectional curvature any $c$-coonvex function is 1-convex; it means that the function $t\mapsto f\circ \gamma(t)+\tfrac{t^2}2$ is convex for any unit-speed geodesic $\gamma$.