I am wondering if there is a simple closed form solution to the constrained proximal mapping problem:
$$ \operatorname*{argmin}_{\beta: \|\beta\|_2 \leq 1} \frac{1}{2\mu }\|X - \beta\|_2^2 + \|\beta\|_1,$$ where $\|a\|_1 = \sum_{i=1}^p |a_i|$. Intuitively, I would think that the solution is the projection of the unconstrained solution (i.e., soft thresholded solution) onto the unit sphere, but I am having difficulty proving this. Perhaps my intuition is wrong here, or I am overlooking a simple property of proximal operators.
Any tips on a direction for proof or papers for reference would be appreciated.