Proximal operator of conjugate function

Question

Asumme $f$ is closed convex function and $f^*$ is the conjugate function. Domain of $f^*$ is $(0,1)$, otherwise $f^*$ is $\infty$. If we directly compute $\textrm{prox}_{f^*}(x) $, the result will be in the domain of $f^*$. From Moreau’s identity we know that \begin{equation}\label{1} \textrm{prox}_{f^*}(x) = x - \textrm{prox}_{f}(x). \end{equation} If we use this equation, is it true that the result is also in the domain of $f^*$?

Answer 1

Moreau's identity always holds for proper convex lower-semicontinuous functions (sometimes called closed convex, depending on the book; I'm assuming these are the hypotheses you mean). For any element of a real Hilbert space $x\in\mathcal{X}$, \begin{equation} y=\textrm{prox}_{f^*}x=(\textrm{Id}+\partial f^*)^{-1}x\;\;\Leftrightarrow\;\;x\in y+\partial f^*y\;\;\Rightarrow \partial f^*(y)\neq\varnothing, \end{equation}

and since $y=\textrm{prox}_{f^*}x\in\{x\in\mathcal{X}\, |\, \partial f^*(x)\neq\varnothing\}\subset\textrm{dom} f^*$ you're done.