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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 $f_1$?

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  • $\begingroup$ What is $A$, the set in which $x$ varies when the supremum is taken? If it is the whole domain of $f$, then $\forall y >1: f_1^*(y) = \infty$ $\endgroup$ – Ingix Mar 14 '17 at 11:36
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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

$$f_1^{\ast}(y)~=~\left\{\begin{array}{rcl} 0&\text{for}& |y| \leq 1,\cr \infty&\text{for}& |y| > 1.\end{array} \right.$$

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