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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.

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  • $\begingroup$ If $x_n=f(w_n)\to a\not= f(w) = x$ for some sequence $w_n\to w$, then $w(a)<w(x)$ by uniqueness of the maximum, which immediately leads to a contradiction because we also have that $w_n(x_n)\ge w_n(x)$. $\endgroup$ Jul 29, 2014 at 21:56

1 Answer 1

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  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.

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