I reading a book which is convex optimization (Stephen Boyd). I have a question in the section "Operations that preserve convexity". Particularly, I said that
If $f$ is convex in (x,y), and $C$ is a convex nonempty set, then the function $$g(x) = \inf_{y \in C} f(x,y)$$ is convex in $x$, provided $g(x) > -\infty$ for all $x$.
After reading that , I tried to solve an exercises in a book
The infimal convolution of two functions $f$ and $g$ on $\mathbf{R}^n$ is defined as $$ h(x)=\inf _y(f(y)+g(x-y)) $$ Prove that if $f, g$ are convex, then $h$ is convex.
Although I know that there are many proofs of the convexity of the infimal convolution available on the internet, I am still posting this topic to discuss it more clearly because they approach it by using other theorems.
Here is my attempt:
We have $t(x, y) = f(y) + g(x - y)$, which is convex because both $f$ and $g$ are convex. Now, let $x \in \mathbb{R}^n$, and I need to prove that $h(x) > -\infty$. However, I am unsure about how to proceed with this. I am concerned that the book might have omitted the assumption that $f$ and $g$ are PROPER convex functions. Have I made any mistakes?
Furthermore, if I suppose that $f$, $g$ should be PROPER, convex. Then, can I continue my proof in the following way:
Fixed $x \in \mathbb{R}$, we have $f(y) + g(x-y) > -\infty$ for all $y \in \mathbb{R^n}$. Then $h(x) > -\infty$ forall $x \in \mathbb{R^n}$.
Thank you a lot for your help.
Even if $f$ and $g$ are proper, the infimal convolution might be $-\infty$ everywhere: Consider $f(x) = x$ and $g(x) = -x$.