Does anyone know reference for a theorem of the following sort:
Proposition: Let $K \subset\mathbb {R}^n$ be a compact convex set, and assume that
$$f(w):=\operatorname{argmax}_{x\in K}w(x) $$ is unique for each nonzero linear functional $w:\mathbb{R}^n\rightarrow \mathbb{R}$. Then the function $f$ is continuous at nonzero $w$.
?
I don't want a proof, just a citeable reference or the name of this theorem or some generalization.