Find the effective domain of $g$, and prove that $g$ is convex.

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.

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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.