Let $V$ be a proper nontrivial closed vector subspace of $\mathbb{R}^N$, and consider the $1$-ball $B_1=\{x\in\mathbb{R}^N\, |\, \|x\|_1\leq \gamma\}$ for some $\gamma\in\mathbb{R}_{++}$.
Is there a closed-form for computing the projection onto $B_1\cap V$?
I know this can be solved numerically, but I'm curious about a closed-form solution. For the $2$-ball $B_2=\{x\in\mathbb{R}^N\,|\,\|x\|_2\leq\gamma\}$, it follows from section 7 in this paper that $P_{B_2\cap V}=P_{B_2}\circ P_V$. Are y'all aware of any similar results for the $1$-norm (or any other norms, for that matter)?
This graph might he helpful for visualization for the case of $\mathbb{R}^2$.