Moreau envelope of the Euclidean norm ($f_{\tau }(x)=\{\begin{array}{ll}\Vert x{\Vert }_2^2/(2\tau ),& \Vert x{\Vert }_2\le \tau ,\\ \Vert x{\Vert }_2-\tau /2,& \Vert x{\Vert }_2>\tau .\end{array}$)

For $r=\Vert x{\Vert }_2$, the Moreau envelope of the Euclidean norm is $r^2/(2\tau )$ when $r\le \tau$ and $r-\tau /2$ otherwise. Its gradient is $x/\tau$ in the core and $x/r$ outside. The approximation error is between zero and $\tau /2$.