Moreau decomposition

For $t>0$, the generalized Moreau decomposition is ${\mathrm{prox}}_{tf}(y)=y-t{\mathrm{prox}}_{t^{-1}{f}^{\ast }}(y/t)$, where $f^{\ast }$ is the convex conjugate.

For a nonempty closed convex set $C$, the Moreau decomposition and ${\sigma }_C^{\ast }={\iota }_C$ give ${\mathrm{prox}}_{t{\sigma }_C}(y)=y-t{\Pi }_C(y/t)$.