= Solution
For the <Euclidean norm>, rotational symmetry and minimizing the radial objective give the <radial soft thresholding> formula
$$
\operatorname{prox}_{\tau\|\cdot\|_2}(x)=
\begin{cases}
0,&\|x\|_2\leq\tau,\\
\left(1-\dfrac\tau{\|x\|_2}\right)x,&\|x\|_2>\tau.
\end{cases}
$$
Indeed, if $r=\|x\|_2$, an optimal $u$ is a nonnegative scalar multiple of $x$; minimizing $s+(r-s)^2/(2\tau)$ over $s\geq0$ gives $s=(r-\tau)_+$. The <gradient of a Moreau envelope> is consequently
$$
\boxed{\partial f_\tau(x)=
\begin{cases}
\{x/\tau\},&\|x\|_2\leq\tau,\\
\{x/\|x\|_2\},&\|x\|_2>\tau.
\end{cases}}
$$
At $x=0$ this is $\{0\}$, and the two expressions agree when $\|x\|_2=\tau$. The <Moreau envelope of the Euclidean norm> itself is
$$
f_\tau(x)=
\begin{cases}
\|x\|_2^2/(2\tau),&\|x\|_2\leq\tau,\\
\|x\|_2-\tau/2,&\|x\|_2>\tau.
\end{cases}
$$
Thus a quadratic core replaces the nondifferentiable tip while the outer gradient remains the normalized radial direction. The <Euclidean norm> has an unbounded <effective domain>, unlike the earlier bounded-domain hypothesis; the preceding argument explicitly shows that this restriction is unnecessary here.
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2016/iii/paper-325-moreau-norm.png]
{title=The absolute value and its Moreau envelope with tau equal to one, together with the continuous clipped gradient replacing the jump at the origin}
For <convex optimization>, this <Moreau–Yosida regularisation> permits <gradient descent> and other methods for functions with a <Lipschitz gradient>, using a gradient Lipschitz constant $1/\tau$. It approximates the norm uniformly: \b[$0\leq\|x\|_2-f_\tau(x)\leq\tau/2$]. Smaller $\tau$ improves approximation but makes the permitted gradient steps smaller. Minimizing $f_\tau$ alone preserves the minimizers and minimum value of $f$; replacing one term in a larger objective can shift the minimizer, so the smoothing parameter controls that approximation error.
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