Consider a metric space $X$ and a set-valued map $F : X \to \mathbb{R}^{n}$. We define the minimal selection
\begin{equation\*}
m(F(x)) := \arg\min \big\{ \lvert u \rvert : u \in F(x) \big\},
\end{equation\*}
where $\lvert \, \cdot \, \rvert$ denotes the 2-norm.


The minimal selection theorem in question can be stated as follows [e.g. Corollary 9.3.3 in Aubin and Frankowska]:


*Let $F$ be a continuous (upper and lower semicontinuous) set-valued map from a metric space $X$ to $\mathbb{R}^{n}$ with nonempty closed convex images. Then the minimal selection is continuous.*


My question is if there exist analogous theorems with $m$ replaced by
\begin{equation\*}
m'(F(x)) := \arg\min \big\{ f(x,u) : u \in F(x) \big\},
\end{equation\*}
where $f$ is a continuous function such that $f(x, \, \cdot \,)$ is strictly convex on $F(x)$?



Generalization of minimal selection theorem
I think the Berge Maximum theorem could be applicable?


<https://en.wikipedia.org/wiki/Maximum_theorem>


See page 116;
[C. Berge, Topological Spaces: including a treatment of multi-valued
functions, vector spaces, and convexity. CC, 1997]


