Let g(·，·):$\mathbb{R}^{n+m}\rightarrow \mathbb{R}$ be a convex function. Let Z be a nonempty compact convex subset of $\mathbb{R}^m$. The function $f(x)=min_{z\in Z}g(x,z)$. It is easy to prove that $f(x)$ is a convex function.
Moreover, I want to show that $conv\{\partial_x g(x,z')|z'\in \mathcal{A}(x)\}\subseteq\partial f(x)$, where $\mathcal{A}(x)=\{z'\in Z:g(x,z')\leq g(x,z),\forall z\in Z\}$, utilizing the optimality condition of $z'\in \mathcal{A}(x)$ for convex problem.
We know that for a convex program $\min_{x\in \mathcal{S}} f(x)$ with a convex function $f(x)$ and a convex set $\mathcal{S}$ if $\bar{x}$ is a local minimizer, $\exists \xi \in \partial f(\bar{x}),\xi^T(x-\bar{x})\geq 0, \forall x\in \mathcal{S}$. But I have no idea about how to connect it with the above problem since the above conclusion should hold for all x not only the optimizer.
There is another idea that we just need to prove that $\partial f(x)$ is a convex set since $conv\{\partial_x g(x,z')|z'\in \mathcal{A}(x)\}=conv(\partial f(x))$?