I am currently studying the concavity of the following function $$ g(x)=\inf_{z\in Z(x)} f(x,z), $$ where for each $z$, $x\to f(x,z)$ is a concave function, locally bounded uniformly in $z$, and $Z(x)$ is a non-empty compact set at each $x$.
If $Z$ is independent of $x$, then it is clear that $g$ is concave in $x$. I was wondering whether there is any existing result asserting that under certain assumptions of $Z$ (maybe regularity under certain metric), $g$ is concave?