Let the convex conjugate function be defined as: $$f^*(y) = \sup_{x \in A} \langle x,y\rangle - f(x)$$ whereas $ \langle x,y\rangle$ denotes the scalar product. I have two functions:
$$f_1: \mathbb R \rightarrow \mathbb R: x\mapsto |x|$$ $$f_2: [-1,1] \rightarrow \mathbb R: x \mapsto 0$$ Now I would like to calculate the convex conjugate of those two functions. For $f_2$ I have: $$f_2^*(y)=\sup_{x\in [-1,1]} xy=|y|$$
What about $f_1$?
The convex conjugate
$$f_1^{\ast}:~~\mathbb{R}\to \mathbb{R}\cup \{\infty\}$$
of the absolute value
$$f_1~:=~|\cdot|:~~\mathbb{R}\to \mathbb{R}$$
is
$$f_1^{\ast}(y)~=~\left\{\begin{array}{rcl} 0&\text{for}& |y| \leq 1,\cr \infty&\text{for}& |y| > 1.\end{array} \right.$$