Convexity of inf convolution

Question

Let $X$ be a compact convex set in a normed linear space, and suppose $f: X \times X \to \mathbb{R}$ is convex, i.e. \begin{equation} f((1-\lambda)x_1 + \lambda x_2, (1-\lambda)y_1 + \lambda y_2) \leq (1-\lambda)f(x_1,y_1) + \lambda f(x_2,y_2). \end{equation} Suppose $g: X \to \mathbb{R}$ is a function which satisfies the relation \begin{equation} g(x) = \inf_{y \in X}\{f(x,y) + g(y)\} \end{equation} Is it necessarily true that $g$ is convex? It is well known that $\inf_{y \in X}f(x,y)$ is convex, but this same type of proof does not seem to work here. On the other hand, the relation for $g$ is incredibly restrictive that one should be able to say something.