by ex.nihil
Last Updated September 21, 2018 22:20 PM

Let $f : \mathbb{R}^n \times \mathbb{R}^m \longrightarrow \mathbb{R}\cup\{+\infty\}$ be a proper and convex function and let $C \subset \mathbb{R}^m$ be a nonempty and convex set.

The function $g : \mathbb{R}^n \longrightarrow \mathbb{R}\cup\{+\infty\}$ is defined by

$$g(x)=\inf_{y\,\in \,C} f(x,y), \;\;\; \forall x\in \mathbb{R}^n.$$

It is assumed that $g$ is a proper function.

$(\text{a})$ Find the effective domain $\text{dom }(g)$.

$(\text{b})$ Prove that $g$ is convex.

If $g(x)=+\infty$, then $f(x,y)=+\infty$ for any $y\in C$, so, if $C$ contains at least two points, $f$ is not proper. Then $g$ is always finite since it is a proper function.

Our $f$ is a pointwise infimum of a family of convex functions, since for any fixed $y\in\Bbb R^m$, the function $\Bbb R^n\ni x\mapsto f(x,y)$ is convex. Such an infimum is always convex, which is a standard fact.

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