I am seeking a projection onto the set defined by
$$\mathcal{C} = \left\{ \mathbf{x} \in \mathbb{C}^m : \left| \mathbf{a}^{T} \mathbf{x} \right| \leq \gamma \right\}$$
where $\mathbf{a} \in \mathbb{C}^m$ is a given complex-valued weight vector of length $m$ and $\gamma$ is some given threshold.
On the other hand, if $\mathbf{b} \in \mathbb{R}^m$ and $\mathbf{y} \in \mathbb{R}^m$ are real-valued vectors, then I can find a projection onto the half-space, e.g. in [1], i.e.,
$$\widetilde{\mathcal{C}} = \left\{ \mathbf{y}: \mathbf{b}^{T} \mathbf{y} \leq \gamma \right\}$$
as
\begin{align*} P_{\widetilde{\mathcal{C}}}\left(\mathbf{y}\right) = \left\{ \begin{matrix} \mathbf{y} + \frac{\left(\gamma - \mathbf{b}^{T} \mathbf{y}\right)}{\left\|\mathbf{b}\right\|_2^2} \mathbf{b}, & \text { if } \mathbf{b}^{T} \mathbf{y} > \gamma; \\ \mathbf{y}, & \text { if } \mathbf{b}^{T} \mathbf{y} \leq \gamma. \end{matrix} \right. \end{align*}
So, do you think there exist a projection operator for this case $\mathcal{C} = \left\{ \mathbf{x}: \left| \mathbf{a}^{T} \mathbf{x} \right| \leq \gamma \right\}$? If yes, I would highly appreciate your help and pointers.
Many thanks in advance
[1] N. Parikh, S. Boyd, Proximal algorithms, Foundations and Trends in Optimization, volume 1, number 3, pages 123-231, 2013.