How to evaluate this nested supremum-infimum formula?

Question

How to evaluate this supremum-infimum, in which $a$ is a positive constant? $$ \sup_x \inf_y (-xy + a |y|) $$ I actually know the answer but I can't really prove it. I believe it is $0$ if $|x| \leq a$ and $-\infty$ otherwise. It comes up in optimization theory. Can someone show me how to prove this?

Answer 1

If $f(x) = \inf_y (a|y|-xy)$, we see that $f(-x) = f(x)$, $f(0) = 0$.

If $|x|\le a$, then $xy \le |xy| \le a|y|$, and so $a|y|-xy \ge 0$, hence $f(x) = 0$.

If $|x|>a$, then $f(x) =f(|x|) \le (a-|x|)y$ for all $y \ge 0$, hence $f(x) = -\infty$.

Consequently, $\sup_x f(x) = 0$.