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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 $F(x)$?

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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]

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    $\begingroup$ While the provided version of the Berge Maximum theorem is not a generalization of the minimum selection theorem as posed in the question, I was indeed unaware of its existence. This seems like a promising avenue of further research. Thank you! $\endgroup$
    – node
    Mar 16, 2020 at 16:13

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