When do minimizers exist?

Question

I'm trying to solve a problem set for my functional analysis course and I'm stuck at the following problem:

Decide if the following problem has a minimizer

Let $g\in C^0([0,1])$. Minimize $\|f-g\|_{L^\infty([0,1])}$ among all $f\in L^\infty([0,1])$ with $$\int_0^1\,f\,dx=0$$

Hint: Consider $$\left|\int_0^1(g-f)\,dx\right|$$

I can't tell you something about what I've done so far, as I don' get the exercise. What are the conditions for such a problem to have a minimizer? What do I need to show? (Maybe there is no general scheme to apply here...?)

I don't have anything in my lecture notes on that..

Thanks in advance!

Answer 1

$$ \Bigl|\int_0^1g\,dx\Bigr|=\Bigl|\int_0^1(g-f)\,dx\Bigr|\le\int_0^1|g-f|\,dx\le\|g-f\|_\infty. $$ Can you think of an $f$ such that

$$ \Bigl|\int_0^1g\,dx\Bigr|=\|g-f\|_\infty? $$