Problem: Prove orthogonal projection onto the $L_{\infty}$ norm ball, for $x \in \mathbb{C}^N$ and $\eta \geq 0$, is
\begin{align} P_{B_{\| \cdot \|_{\infty} \leq \eta}} &= \left[\operatorname{sign}(x_n) \min\left\{ |x_n|, \eta \right\} \right]_{1\leq n \leq N} \\ &= \left[\left(\frac{x_n}{|x_n|}\right) \min\left\{ |x_n|, \eta \right\} \right]_{1\leq n \leq N} . \end{align}
The source of the above result is here http://proximity-operator.net/indicatorfunctions.html. However, there is no reference. So, I wonder how it is proved for the complex-valued. There are some posts that prove orthogonal projection onto the $L_{\infty}$ norm ball for the real-valued data, e.g., Orthogonal Projection onto the $ {L}_{\infty} $ Unit Ball. Does any expert know how to prove it for the complex-valued? Thank you so much in advance.
