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.



Reference request: Continuity of unique maximizer of linear functional on convex set
1. Let $\sigma(w) = \max\_{x\in K} w(x)$ be the support function of set $K$. This function is convex and its subdifferential is exactly your function $f$, i.e. $f(w) = \partial \sigma(w)$.
2. Your assumption that $f(w)$ is unique for all $w \neq 0$ is equivalent to the differentiablity of $\sigma$ on the set of nonzero $w$.
3. We now use the following (which can be found for example in *Convex Analysis and Minimization Algorithms I: Part 1: Fundamentals* by Hiriart-Urrut and Lemaréchal, Remark 6.2.6)



> 
> If a convex function is differentiable on an open set, then it is continuously differentiable on that set.
> 
> 
> 


to conclude that $\sigma$ is continuously differentiable on the region $w \neq 0$, which is the same as saying that $f$ is continuous.


